Ann. Math. Québec https://doi.org/10.1007/s40316-018-0100-y
K1 -congruences for three-dimensional Lie groups Daniel Delbourgo1
· Qin Chao1
Received: 22 August 2017 / Accepted: 21 March 2018 © Fondation Carl-Herz and Springer International Publishing AG, part of Springer Nature 2018
Abstract We completely describe K1 (Z p [[G∞ ]]) and its localisations by using an infinite family of p-adic congruences, where G∞ is any solvable p-adic Lie group of dimension 3. This builds on earlier work of Kato when dim(G∞ ) = 2, and of the first named author and d × Lloyd Peters when G∞ ∼ = Z× p Z p with a scalar action of Z p . The method exploits the classification of 3-dimensional p-adic Lie groups due to González-Sánchez and Klopsch, as well as the fundamental ideas of Kakde, Burns, etc. in non-commutative Iwasawa theory. Résumé Nous décrivons complètement K1 (Z p [[G∞ ]]) et ses localisations en utilisant une famille infinie de congruences p-adiques, où G∞ est un groupe de Lie résoluble de dimension trois. Ce travail s’appuie sur les résultats de Kato lorsque dim(G∞ ) = 2, et du premier auteur d × et Lloyd Peters lorsque G∞ ∼ = Z× p Z p avec une action scalaire de Z p . La méthode exploite la classification des groupes de Lie de dimension trois due à González-Sánchez et Klopsch, ainsi que les idées fondamentales de Kakde, Burns etc. en théorie d’Iwasawa non-commutative. Keywords Iwasawa theory · K -theory · p-adic L-functions · Galois representations Mathematics Subject Classification 11R23, 11G40, 19B28
Contents 1 Introduction . . . . . . . . . . . . . 1.1 Preliminaries . . . . . . . . . . 1.2 The main results . . . . . . . . . 1.3 Some arithmetic examples . . . 2 The general set-up in dimension three
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Qin Chao: To form a part of this author’s PhD thesis.
B 1
Daniel Delbourgo
[email protected] Department of Mathematics, University of Waikato, Hamilton 3240, New Zealand
123
D. Delbourgo, Q. Chao 2.1 Determining the stabilizer of a character on H∞ . 2.2 A “coarse but clean” system of subgroups . . . . 2.3 Maps between the abelianizations of Um,n . . . . 3 The additive calculations . . . . . . . . . . . . . . . . (m,n) . 3.1 The image of under the characters on H∞ 3.2 A transfer-compatible basis for the set Rm,n . . . 4 The multiplicative calculations . . . . . . . . . . . . 4.1 Convergence of the logarithm on Im(σm ) . . . . 4.2 Interaction of the theta-maps with both ϕ and log 4.3 The image of the Taylor-Oliver logarithm . . . . 4.4 A proof of Theorems 1 and 2 . . . . . . . . . . . 5 Computing the terms in Theorems 1 and 2 . . . . . . 5.1 A worked example for Case (II) . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .
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1 Introduction Over the last twenty years, the study of non-commutative Iwasawa theory for motives has progressed rapidly, due to the work of many mathematicians [2,3,6,17–20,23]. Fix an odd prime p, and an infinite algebraic extension F∞ /F of some number field F. We assume that G ∞ = Gal(F∞ /F) is a p-adic Lie group with no element of order p; we further suppose that F∞ contains the cyclotomic Z p -extension F cyc of the base field F. Clearly if H∞ = Gal F∞ /F cyc , then the quotient = G ∞ /H∞ will be isomorphic to an open subgroup of 1 + pZ p , under the p-th cyclotomic character ‘κ F ’. For a motive M with good ordinary reduction at p, the work of Coates et al [6] associates (under the M H (G)-conjecture) a characteristic element ξ M ∈ K 1 Z p [[G ∞ ]]S ∗ , where K 1 (−) denotes the first algebraic K -group, and S ∗ is the p-saturation of the Ore set S := f ∈ Z p [[G ∞ ]] Z p [[G ∞ ]] Z p [[G ∞ ]] f is a finitely-generated Z p [[H∞ ]]-module . The “Non-commutative Iwasawa Main Conjecture” predicts that there exists an element
an ∈ K Z [[G ]] ∗ of the exact form L an = u·ξ with u in the image of K Z [[G ]] ; LM 1 p ∞ S M 1 p ∞ M
for any Artin representation ρ : G ∞ → GL(V ), its evaluation at ρ ⊗ κ Fk should then satisfy k an LM ρκ F = the value of the p-adic L-function L p (M, ρ, s) at s = k,
as the variable k ranges over the p-adic integers. Note that the existence of L p (M, ρ, s) is in most cases still conjectural, although its interpolation properties are easy to describe. Remark The strategy of Burns and Kato [2,20] reduces this conjecture to the following: (1) prove the abelian Iwasawa Main Conjectures for M over all finite layers; (2) describe K 1 Z p [[G ∞ ]]S ∗ via a system of non-commutative congruences; and (3) show that each of the abelian fragments, L p (M, ρ, −), in combination satisfy this system of congruences. There seem to be two approaches to (2), either using congruences modulo trace ideals [1,17,20,21,23], or instead by deriving p-adic congruences [10–12,16,18,19]. Naturally both approaches should be equivalent to one another. To illustrate precisely what is meant by the terminology ‘ p-adic congruences’ above, for the moment suppose that G ∞ is a two-dimensional p-adic Lie group of the form × ∼ G∞ ∼ = Z× p Zp = Fp × Zp
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K1 -congruences for three-dimensional
where = 1 + pZ p , and the first factor Z× p acts on the second Z p via scalar multiplication. Let ϕ : Z p [[]] → Z p [[]], ϕ : γ → γ p denote the linear extension of the p-power map on m
. At integers m ≥ m ≥ 0, we also write Nm ,m : Z p [[ p ]] → Z p [[ p ]] for the norm map. m
× Kato’s Theorem ([19] [8.12]) A sequence ym ∈ m≥0 Z p p ( p) arises from an ele ment in K 1 Z p [[G ∞ ]]S , only if the system of p-adic congruences m m =1
Nm ,m
m
pm
m
ϕ N0,m −1 y0 ym
· ≡ 1 mod p 2m · Z p p ( p) ϕ ym −1 N0,m y0
hold at every integer m ≥ 1. s
Kato has obtained similar congruences when G ∞ is replaced by any of the groups p Z p . His work completely describes the two-dimensional situation, since any non-commutative s torsion-free pro- p-group G with dim(G) = 2 is isomorphic to p Z p for some s ≥ 0. Question. Can the analogue of Kato’s p-adic congruences be proven when dim(G) > 2? Our goal here is to give a positive answer when dim(G) = 3 and G = SL2 (Z p ), SL1 (D p ). We exclude the two insolvable cases as the representation theory is unpleasant, although recent work of Kakde [18] provides hope that an answer for GL2 (Z p ) is not too far away.
1.1 Preliminaries Fix a number field F and a prime number p = 2. We shall assume that F∞ denotes a p-adic Lie extension of F satisfying: (i) Gal(F∞ /F) is a pro- p-group without any p-torsion; (ii) F∞ contains the cyclotomic Z p -extension F cyc of F. The examples we have in mind here are solvable three-dimensional Galois groups arising from algebraic geometry, or alternatively the direct product of a two-dimensional Galois group with a group of diamond operators (in the context of Hida’s deformation theory). We therefore suppose that either (iiia) G∞ = Gal(F∞ /F) where dim Gal(F∞ /F) = 3 and G∞ SL2 (Z p ), SL1 (D p ); or
(iiib) G∞ = Gal(F∞ /F) × W∞ where dim Gal(F∞ /F) = 2 and W∞ ∼ = Zp. In both (iiia) and (iiib), the p-adic Lie group G∞ is three-dimensional and also solvable; in fact G∞ is a semi-direct product of Z p with an abelian subgroup H∞ of Z p -rank two. The following result classifies such groups. Classification Theorem (González-Sánchez and Klopsch [15]) If the pro- p-group G∞ is solvable and torsion-free with dim(G∞ ) = 3, then G∞ must be isomorphic to one of the following possibilities: (I) the abelian group Z p × Z p × Z p ; (II) an open subgroup of the p-adic Heisenberg group, i.e. a group given by the presentation ps γ , h 1 , h 2 : [h 1 , h 2 ] = 1, [h 1 , γ ] = 1, [h 2 , γ ] = h 1 for some s ∈ N0 ;
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D. Delbourgo, Q. Chao
ps ps (III) the group γ , h 1 , h 2 : [h 1 , h 2 ] = 1, [h 1 , γ ] = h 1 , [h 2 , γ ] = h 2 for some s ∈ N; p s p s+r d p s+r p s , [h 2 , γ ] = h 1 h 2 for some s, r ∈ N (IV) γ , h 1 , h 2 : [h 1 , h 2 ] = 1, [h 1 , γ ] = h 1 h 2 with d ∈ Z p ; ps d p s p s+r (V) γ , h 1 , h 2 : [h 1 , h 2 ] = 1, [h 1 , γ ] = h 2 , [h 2 , γ ] = h 1 h 2 where s, r ∈ N0 and d ∈ Z p , such that either s ≥ 1, or instead r ≥ 1 and d ∈ pZ p ; p s+r ps (VI) either one of the groups: (a) γ , h 1 , h 2 : [h 1 , h 2 ] = 1, [h 1 , γ ] = h 2 , [h 2 , γ ] = h 1 or p s+r t ps (b) γ , h 1 , h 2 : [h 1 , h 2 ] = 1, [h 1 , γ ] = h 2 , [h 2 , γ ] = h 1 where s, r ∈ N0 such that s + r ≥ 1, and t ∈ Z× p is not a square modulo p. Let = γ z z ∈ Z p where γ is as in the previous theorem (if G∞ = Gal(F∞ /F) satisfies condition (iiia) above, we shall identify its quotient Gal(F cyc /F) ∼ = Z p with ). One defines a decreasing sequence of normal subgroups for G∞ by m
Um := p H∞ at each m ≥ 0.
Recall from [24, Prop 25], every irreducible G∞ -representation with finite image is of the ∞ ab p m → Q× . ∞ form ψ ⊗ IndG p Um (χ) for some m ≥ 0, with characters χ : Um → μ p and ψ : If G is a pro- p-group, then we write (G) = lim P Z p [G/P] for its Iwasawa algebra ← − where the inverse limit runs over open subgroups P G. If O contains Z p as a subring then O (G) := (G) ⊗Z p O. Lastly for a canonical Ore set S , we use (G)S and (G)S ∗ for the localisation of (G) at S , and at its p-saturation S ∗ = n≥0 p n S , respectively. Remark Let
us write NUm : (G∞ ) → (Um ) for the norm mapping on Iwasawa algebras. If Um , Um denotes the commutator subgroup of Um , there is a commutative diagram m × NUm (−) mod [Um ,Um ] χ∗ K 1 (G∞ ) −→ K 1 (Umab ) −→ Oχ p m≥0
K 1 (G∞ )S
NUm (−) mod [Um ,Um ]
−→
NUm (−) mod [Um ,Um ]
−→
χ∗
−→
m≥0 ρχ
⏐
m × Oχ p ( p)
→
K 1 (G∞ )S ∗
K 1 (Umab )S
m≥0
⏐
m≥0 ρχ
⏐
→
⏐
χ∗ m × K 1 (Umab )S ∗ −→ Quot Oχ ( p ) m≥0 ρχ
m≥0
where the vertical arrows are induced from the inclusions (G∞ ) → (G∞ )S → (G∞ )S ∗ , and the right-most products range over irreducible non-isomorphic G∞ -representations. One can then define three separate theta-maps ∞,χ , ∞,χ ,S and ∞,χ,S ∗ by composing (respectively) the first, second and third rows in the above diagram, so that m χ × Oχ p , ∞,χ : K 1 (G∞ ) −→
ρχ
∞,χ,S : K 1 (G∞ )S −→ and
∞,χ ,S ∗ : K 1 (G∞ )S ∗ −→
ρχ
ρχ
123
m χ × Oχ p ( p) m χ × Quot Oχ ( p ) .
K1 -congruences for three-dimensional
The Main Goal. To describe the images of ∞,χ , ∞,χ,S and ∞,χ ,S ∗ by using a family m χ × of p-adic congruences linking together the abelian fragments yρχ ∈ Quot Oχ ( p ) . Note that Case (I) is devoid of any content since G∞ ∼ = × H∞ is abelian, in which case K 1 (G∞ ) = K 1 ( × H∞ ) ∼ = ( × H∞ )× by Morita invariance. Hence one may ignore Case (I) completely, since there are no non-abelian congruences to consider there.
1.2 The main results In order to describe the congruences in each of the non-empty Cases (II–VI), we first need some means to keep track of those Artin representations induced from characters on H∞ . If χ is a finite order character on H∞ then χ extends naturally to Stab (χ) H∞ , hence ∞ ρχ := IndGStab (χ) (χ )H∞
is an irreducible G∞ -representation of dimension p mχ , where mχ = ord p : Stab (χ) . In all cases ∈ {II,III,IV,V,VI}, one constructs characters χ1,n , χ2,n : H∞ → μ p∞ via √ √ y y χ1,n h 1x h 2 = exp 2π −1 x/ p n and χ2,n h 1x h 2 = exp 2π −1 y/ p n for each x, y ∈ Z p . In particular, χ1,n and χ2,n together generate a basis for Hom(H∞ , μ pn ). Case (II). For simplicity, let us initially assume we are in Case (II). Then for each character a · χb x y ∗ n χ = χ2,n 1,s+m and group element h = h 1 h 2 ∈ H∞ , one defines eχ ,h ∈ Z[μ p ] by the formula
χ −1 (h) · p max{0,m − ord p (b)} if p m | by ∗ eχ ,h :=
0 if p m by. m χ × Theorem 1 If we are in Case (II), then a sequence yρχ ∈ ρχ Oχ p ( p) belongs to the image of ∞,χ ,S only if m
n−m p
m =0
a=1
s+m p
b = 1, p b if m > 0
Nmχ ,m
∗ eχ,h ϕ N0,mχ −1 y1 yρ χ · ϕ yρ χ p N0,mχ y1
a ·χ b χ =χ2,n 1,s+m
m
≡ 1 mod p s+m+n+ord p (y) · Z p p ( p)
(1)
y
for all integer pairs m, n ≥ 0 with m ≤ n − s, and at every choice of h = h 1x h 2 ∈ H∞ with x ∈ {1, . . . , p n } and y ∈ {1, . . . , p m }. ∗
We should point out that, a priori, it is not clear whether the p-adic power Nmχ ,m (. . . )eχ,h above should even exist, as the exponent eχ∗ ,h ∈ Z[μ pn ] is frequently not a rational integer! Remarks (i) For any function f (X ) ∈ 1 + p · OC p [[X ]], and provided that s ∈ C p is chosen to lie inside the disk s p < p ( p−2)/( p−1) , the p-adic power series defined as f (X )s := exp p s log p f (X ) converges to an element of 1+ p · OC p [[X ]]. In particular, if s ∈ Z then f (X )s coincides with the standard definition of the s-th power.
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(ii) Furthermore, this construction extends after localisation at the multiplicatively closed set OC p [[X ]] − p · OC p [[X ]], i.e. if f (X ) ∈ 1 + p · OC p [[X ]]( p) then f (X )s ∈ 1 + p · OC p [[X ]]( p) . (iii) Although not explicitly stated, it is nevertheless inbuilt into Theorem 1 that each of the ϕ(N0,m −1 (y1 )) yρ m fractions ϕ(yρχ ) · N0,mχ (y1 ) belongs to the multiplicative group 1+ p· Oχ [[ p ]]( p) . χ
χp
∗
In lieu of this discussion, one deduces that each term Nmχ ,m (. . . )eχ,h in the above them orem exists as a well-defined element of the multiplicative group 1 + p · OC p [[ p ]]( p) . Cases (III)–(VI). Let us now instead suppose we are in Case () with ∈ {III, IV, V, VI}. We define a non-negative integer , p by the rule ⎧ ⎪ if = (III) or (IV) ⎨0 , p = ord p (d) if = (V) ⎪ ⎩ r + ord p (t) if = (VI). It will be shown (in Proposition 7) that the abelianization of Um yields the tricyclic group Um
Umab := Um , Um
m ∼ = p × C ps+m+, p × C ps+m
where Cd denotes the cyclic group of order d. Note that the commutator [Um , Um ] is actually a subgroup of H∞ , while acts on Umab m through the finite quotient / p ; we can then partition (m)
H∞
:=
H∞
Um , Um
∼ = C ps+m+, p × C ps+m
(m) into a finite disjoint union of its -orbits. Similarly, the dual group Hom H∞ , C× also m has an action of / p ; let ‘Rm ’ denote a set of representatives for its -orbits. (m) (m) For each orbit h = γ − j hγ j j ∈ Z/ p m Z , h ∈ H∞ and character χ : H∞ → C× , ∗ we generalise the definition of eχ ,h by computing the trace of h over the orbits of χ: eχ∗ , = Tr( Indχ ∗ ) h := h
(χ )−1 (h).
χ ∈{χ g | g∈}
In fact, it is easy to check that eχ∗ , depends only on the image of χ within the set Rm and h
on the orbit h generated by h, but not on the individual choices of χ and h. Although these quantities might seem abstract, they are all computable (see Lemma 35). mχ × Theorem 2 If we are in Cases (III)–(VI), then a sequence yρχ ∈ ρχ Oχ p ( p) belongs to the image of ∞,χ,S only if χ ∈Rm
Nmχ ,m
∗ eχ, yρ χ ϕ N0,mχ −1 y1 · ϕ yρ χ p N0,mχ y1
m
≡ 1 mod p 2s+3m+, p −ord p (# ) · Z p p ( p)
(m) for every m ≥ 0, and over all -orbits inside the group H∞ ∼ = C ps+m+, p × C ps+m .
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(2)
K1 -congruences for three-dimensional
mχ × Note in both of these theorems, if one additionally knows that yρχ ∈ ρχ Oχ p , the modified statement should read: ‘ yρχ ∈ Im ∞,χ if and only if the same congruences m m in (1), (2) hold after replacing p • · Z p [[ p ]]( p) with its unlocalised version p • · Z p [[ p ]]’. We also remark that Burns and Venjakob [3, Prop 3.4] have constructed a splitting K 1 (G∞ )S ∗ ∼ = K 1 (G∞ )S ⊕ K 0 F p [[G∞ ]] so one can reduce the existence of elements in K 1 (G∞ )S ∗ to those in K 1 (G∞ )S , combined with a precise growth formula for the μ-invariant of the individual yρχ’s.
1.3 Some arithmetic examples Before explaining the strategy to prove our two main theorems, we first discuss some applications to non-commutative Iwasawa theory that arise from these K 1 -congruences. Totally real extensions. Let us initially suppose that F is a totally real field, and further: • F∞ = n≥1 Fn is a union of totally real fields; • only finitely many primes of F ramify inside F∞ /F; cyc of F; • F∞ contains the cyclotomic Z p -extension F • the cyclotomic μ-invariant of F e2πi/ p vanishes. (m) We denote by the primes ramifying inside F∞ /F. One also defines unique F to bethe(m) m cyc cyc ∼ extension of degree p contained in F , so that = Gal F /F = limm Gal F /F . ← − Let G∞ = Gal F∞ /F , and write κ F : → Z× p for the p-th cyclotomic character. By seminal work of Burns, Kakde and Ritter-Weiss [2,17,23], there exists an element ζ F∞ /F ∈ K 1 (G∞ )S ∗ such that, at any Artin representation ρ : G∞ → GL(V ), one has ζ F∞ /F ρκ Fk = L (ρ, 1 − k)
for each k ∈ N satisfying k ≡ 0 (mod[F(μ p ) : F]). By deforming the k-variable p-adically, the above values interpolate to the Iwasawa function L p, (ρ, −) : Z p → Q p constructed by Cassou-Noguès and Deligne-Ribet [4,14]. Corollary 3 Let F∞ /F be an infinite solvable Lie extension as above, with dim(G∞ ) = 3. ∞ If the representation ρχ = IndGStab (χ) has dimension equal to p mχ say, then write (χ )H∞ D-R ρ p mχ ) × for the unique element satisfying L p, χ ∈ Quot Oχ ( D-R ρχ = L p, ρχ , 1 − k for allk ∈ Z p . κ Fk ◦ L p, D-R ρ . (a) If we are in Case (II), then the system of congruences (1) holds for yρχ = L p, χ D-R ρ . (b) In Case () with ∈ {III,IV,V,VI}, the congruences (2) hold for yρχ = L p, χ m χ × D-R ρ Proof Note that the infinite sequence L p, ∈ ρχ Quot Oχ ( p ) coincides χ with ∞,χ ,S ∗ ζ F∞ /F , as they both interpolate the same L-values. Therefore the necessity of the congruences (1) and (2) follows directly from Theorems 1 and 2, respectively. Let us now digress momentarily, and assume we are given a congruence of the form F(X ) ≡ 1 mod p v · Z p [[X ]]( p) with F, G ∈ OC p [[X ]] and v ≥ 1. G(X )
123
D. Delbourgo, Q. Chao F(X ) v R(X ) Then G(X ) = 1 + p · T (X ) for some R, T ∈ Z p [[X ]] where the μ-invariant of T equals zero. It follows that F · T = G · (T + p v · R), and one works out that
μ(F) = μ(F · T ) = μ(G) + μ(T + p v · R) = μ(G) + 0, v i.e. μ(F) = μ(G). Also F = G + p ·RG ∈ OC p [[X ]] so that T RG, whence F ≡ G T ( mod p v ). Certainly if μ(F) = μ(G) = 0, then the leading terms of F and G are congruent mod p v . However even if μ(F) = μ(G) > 0, their leading terms must still be congruent modulo p v , as one can repeat the above argument with F˜ = p −μ(F) · F and G˜ = p −μ(F) · G instead. F(X ) v v Conclusion: If G(X ) ≡ 1 mod p · Z p [[X ]]( p) , the leading terms of F, G agree modulo p . We are going to apply this to the congruences (1) and (2) at the trivial orbit = { id}: specifically, F will denote the numerator of (1) and (2) while G will be the denominator, so F(X ) v p m − 1, and v = s + 2m + n when = II that G(X ) ≡ 1 mod p · Z p [[X ]]( p) with X = γ whilst v = 2s + 3m + , p when = II. To individually describe the leading terms, if r (ρ, x0 ) = orderx=x0 L p, (ρ, x) then L (ρ, 1 − k) if r (ρ, 1 − k) = 0 ( p) −r (ρ,1−k) L (ρ, 1 − k) := lim x→1−k x · L p, (ρ, x) if r (ρ, 1 − k) > 0 yields the p-adic residue of L p, (ρ, x) at the non-positive critical value x = 1 − k. Notations: (m ) (i) At integers m ≥ m ≥ 0, let us define rm ,m = Ind FF (m) (1) to be the regular represen
tation for Gal F (m) /F (m ) . (m ) (ii) Furthermore, we shall write r0,m as an abbreviation for Ind F (m −1) ψ p ◦ rm ,m (m ) , F F where ψ p is the p-th Adams operator (strictly speaking ψ p only acts on the trace of a virtual representation, but the abuse of notation makes sense in the context of ζ -functions). (mχ ) (m) (m) (iii) Lastly set ρχ := Ind FF (m) χ F (m) and ρχ p := Ind F (mχ −1) ψ p ◦ Ind FF (m) χ F (m) . F Theorem 4 Let F∞ /F be as above, with dim(G∞ ) = 3 and also ζ F∞ /F ∈ K 1 (G∞ )S . (a) If we are in Case (II), then for every m, n, k ∈ N:
≡
m
n−m p
m =0
a=1
m
n−m p
m =0
a=1
s+m p
b = 1, p b if m > 0
s+m p
b = 1, p b if m > 0
( p) (m ) pmχ ( p) L ρχ(m) , 1 − k · L r0,mχ , 1 − k
( p) pmχ ( p) (m) L ρχ p , 1 − k · L r0,m , 1 − k
a ·χ b χ =χ2,n 1,s+m
a ·χ b χ =χ2,n 1,s+m
modulo p s+2m+n . (b) In Case () with ∈ {III,IV,V,VI}, for every m, k ∈ N: ( p) ( p) (m ) p m χ L ρχ(m) , 1 − k · L r0,mχ , 1 − k χ ∈Rm
≡
χ ∈Rm
123
( p) p m χ ( p) (m) L ρχ p , 1 − k · L r0,m , 1 − k
mod p 2s+3m+, p .
K1 -congruences for three-dimensional
Because p-adic zeta-functions of totally real fields do not vanish at odd negative integers, a nice consequence is that whenever k ≡ 0 (mod [F(μ p ) : F]), these congruences actually involve bona fide complex zeta-values, not simply their p-adic residues. Heisenberg extensions. Let us now suppose we are in Case (II) with the parameter s ≥ 0, in which case G∞ is an open subgroup of the Heisenberg group, i.e. ⎛ ⎞ 1 Zp Zp
Z p ⎠ where H3 (Z p ) : G∞ = p s . G∞ H3 (Z p ) := ⎝ 0 1 0 0 1 In an unpublished preprint [20], Kato derives different but equivalent congruences to (1), as ideal congruences in the group algebras associated to finite sub-quotients of H3 (Z p ). Thus Theorem 4(a) gives a concrete description for the most basic of these ideal relations, as a congruence modulo p s+2m+n connecting the special values of Artin L-functions. 1/ p ∞ 1/ p ∞ , q2 False-Tate extensions. Fix s ≥ 1. We set F = Q(μ ps ) and F∞ = Q μ p∞ , q1 where q1 , q2 > 1 are distinct p-power free integers satisfying gcd( p, q1 q2 ) = gcd(q1 , q2 ) = 1. Then G∞ = Gal F∞ /F is a three-dimensional pro- p-group, corresponding to Case (III) in the Classification Theorem (note that F∞ is not a union of totally real fields so there is no element ζ F∞ /F ∈ K 1 (G∞ )S ∗ available, and therefore no Iwasawa Main Conjecture can be formulated for Tate motives here). Now if s = 1, the congruences (2) specialise down to yield the congruences labelled (1.1)m,h and (1.2)m in [10, p3]. If E /Q denotes a semistable elliptic curve with good ordinary mχ
reduction at p, then p-adic L-functions L p (E, ρχ ) ∈ p 1/ p interpolating the algebraic part of L { pq1 q2 } (E, ρχ , 1) have been constructed in Theorem 1.5 of op. cit. Furthermore, there are three ‘first layer congruences’ to check for each tuple (E, p, q1 , q2 ). These were verified numerically for the elliptic curves 11a3, 77c1, 19a3 and 56a1 using MAGMA at the primes p = 3, 5 and at small values of q1 and q2 , in Sect. 6 of op. cit. On the algebraic side, let us further assume that q1 and q2 are both chosen to be primes of non-split multiplicative reduction for E, such that ! " l ( p−1)/2 × (−1) = −1 p
l| cond(E), l =q1 ,q2
− p
denotes the Legendre symbol at p. Then if the cyclotomic λ-invariant of E/Q(μ p∞ ) equals one and if Sel p∞ (E/F∞ )∧ belongs to the category MH∞ (G∞ ), Sel it is shown in [9, Corollary 2.6] that rankZ E(Fn ) = p 2n−1 or p 2n , 1/ p n 1/ p n . provided the p-Sylow subgroup of III(E/Fn ) is finite at each layer Fn = Q μ pn , q1 , q2 Alternatively, by studying the λ-invariants of each χ-part Sel p∞ (E/Fn (μ p∞ ))∧ ⊗Z p ,χ Oχ using the congruences in Theorem 2, one can produce the same estimate for the rank (current work of the first named author [13]). √ Heegner-type extensions. Consider an imaginary quadratic field k = Q −D and let us suppose k∞ denotes its Z2p -extension, so that Gal(k∞ /k) ∼ = × H1,∞ where H1,∞ is the Galois group of the anticyclotomic Z p -extension of k. For any choice of odd prime q = p √ ∞ with q D, one may set F = Q −D, μ p and F∞ = k∞ μ p , q 1/ p , in which case G∞ := Gal(F∞ /F) ∼ = × H1,∞ H2,∞ . = H1,∞ × H2,∞ ∼ where
p∞
123
D. Delbourgo, Q. Chao
Here h 1 acts trivially on H2,∞ = h 2 = Gal F∞ /k∞ (μ p ) , while γ acts on h 2 through multiplication by 1 + p (we must therefore be in Case (V) with s = d = 0 and r = 1). Let E /Q be a semistable elliptic curve with ordinary reduction at p, split multiplicative reduction at q, and with non-split multiplicative reduction at all other primes dividing the × conductor of E. We also suppose that q generates Z/ p 2 Z so that q is inert in Q(μ p∞ ), and that the various Heegner conditions (DT1)–(DT7) described in [9, Sect 2.4] hold. Then it is shown in Proposition 2.14 of op. cit. that for n 0, ! " 2 p2 + 2 p + 1 2n p · 1− ≤ rankZ E(Fn ) ≤ p 2n + 4 ( p + 1)3 with no hypotheses whatsoever on the finiteness of III(E/Fn )[ p ∞ ]. The upper bound essentially comes from a growth formula for the λ-invariant of Sel p∞ (E/Fn (μ p∞ ))∧ as n becomes large. In fact if one the congruences (2), this exploits yields another way to obtain the upper bound on rankZ E(Fn ) , and establishes finer bounds on the χ-part of E(Fn ). However the lower bound relies heavily on the properties of Heegner points, following the same approach as Darmon and Tian [8] in dimension 2. p n -division√ fields of CM curves. Let E /Q be an elliptic curve with complex multiplication by k = Q −D , and select a good ordinary prime p = 2 for E which splits inside √ √ n Z −D . If one takes F = Q −D, μ p , Fn = Q E[ p n ], q 1/ p and F∞ = n≥1 Fn for an auxiliary prime q not dividing cond(E), then G∞ := Gal(F∞ /F) corresponds to Case (V) with s = d = 0 and r = 1 again. By using the congruences (2) to study the λ-invariants of Sel p∞ (E/Fn )∧ , one can bound the rank of E(Fn ) from above by p 2n if the cyclotomic λ-invariant is one. Whilst Heegner points are no longer useful here, a lower bound on the Z-rank of E(Fn ) of the form c p × p 2n (with c p = 0 and c p ∼ 1 if p 0) should still be feasible, if one exploits the non-triviality of the Euler system of elliptic units in place of the Heegner points. Here is a brief plan of the article. In Sect. 2 we begin by choosing an appropriate system of subgroups with which to define our theta-map. The choice we make differs from that made in [17] - ours is a coarser system than Kakde’s choice, yet better suited to the specific representation theory of G∞ . We then write down bases for each piece of the image of the theta-map, and also introduce auxiliary homomorphisms Ver and π which allow us to pass between adjacent subgroups in this directed system. The additive component of the proof is contained in Sect. 3, where we describe the image of the additive theta-map through its special values at Artin representations ρχ . We next formulate four conditions (C1)–(C4), which are just strong enough to determine whether or not an element lies in the image of this homomorphism. In Sect. 4, we pass from the additive to the multiplicative world by means of the TaylorOliver logarithm; for those familiar with the details of [7, p 79–123], under this logarithm the conditions ‘(M1)–(M4)’ transform into our additive conditions (C1)–(C4). Because our subgroup system is coarser than in op. cit., the proof of the converse statement “(C1)– (C4) ⇒ (M1)–(M4)” is far from immediate and occupies much of this article. Finally in Sect. 5, we develop an algorithm to compute the quantities Rm , eχ∗ , , # in Theorem 2 explicitly, using Case (II) as a worked example to trial the algorithm.
2 The general set-up in dimension three We shall begin by reviewing the representation theory behind these semi-direct products. Let us first observe that the subgroup H∞ = H1,∞ × H2,∞ ∼ = Z p × Z p is generated by
123
K1 -congruences for three-dimensional
h 1 = (1, 0)T and h 2 = (0, 1)T topologically. The action of each g = γ z ∈ on an arbitrary y element (x, y)T = h 1x h 2 ∈ H∞ can be described through a 2 × 2-matrix of the form I2 + M: ! " z x y for all g = γ z ∈ γ z (x, y)T = γ −z h 1x h 2 γ z = I2 + M y ! " 1 0 is the identity, and M ∈ Mat2×2 Z p is topologically nilpotent. where I2 = 0 1 Applying the Classification Theorem for G∞ , the matrix M equals " ! s " ! " ! ! " ! s " p 0 ps 0 ps 0 ps 0 p s+r p , , and , (3) 0 ps 0 0 p s+r d ps p s d p s+r p s+r t 0 in Cases (II), (III), (IV), (V) and (VI) respectively (note in Case (VIa) we have set t = 1).
2.1 Determining the stabilizer of a character on H∞ Note that each element g ∈ acts naturally on χ ∈ Hom(H∞ , μ p∞ ) by sending χ → g ∗χ, where g ∗ χ(h) := χ(g −1 hg) for all h ∈ H∞ . The -stabilizer of χ is given by the subgroup y y y Stab (χ) := g ∈ χ g −1 (h 1x h 2 )g = χ h 1x h 2 for all h = h 1x h 2 ∈ H∞ . e1 e2 Proposition 5 If χ = χ1,n × χ2,n : H∞ μ pn is a surjective character, then
˜ χ} : Stab (χ) = p mχ where mχ := max{0, m
and, using the case-by-case description in the Classification Theorem, one has respectively: ˜ χ = n − s; ˜ χ = n − s; ˜ χ = n − s − ord p(e1 ); (III) m (IV) m (II) m re ) ; ˜ χ = n − s − min ord p (e2 ) + ord p (d), ord (V) m (e + p and p 1 2 ˜ χ = n − s − min r + ord p (e2 ), ord p (e1 ) . (VI) m √ Proof Firstly, let us denote by ζ pn the primitive p n -th root of unity exp(2π −1/ p n ). " ! i i 1 ps y x+ p s+i y y Case (II). Here I2 + M = , so that γ − p (h 1x h 2 )γ p = h 1 h 2 . Consequently 0 1 x+ ps+i y y e1 x+ ps+i y y e2 i i y e x+(e2 +e1 × p s+i )y χ γ − p (h 1x h 2 )γ p = χ1,n h 1 h 2 × χ2,n h 1 h2 = ζ p1n y e x+e y equals χ h 1x h 2 = ζ p1n 2 for all x, y ∈ Z, if and only if e1 × p s+i ≡ 0 ( mod p n ). ! " 1 + ps 0 Case (III). Here I2 + M = with repeated eigenvalue λ III,± = 1 + p s , 0 1 + ps and it follows that i i i s pi y y (e x+e y)×(1+ p s ) p χ γ − p (h 1x h 2 )γ p = χ(h 1x h 2 )(1+ p ) = ζ pn1 2 . i
i
However (1 + p s ) p ≡ 1 (mod p s+i ) but (1 + p s ) p ≡ 1 (mod p s+i+1 ), in which case i i y y e x+e y − p χ γ (h 1x h 2 )γ p equals χ(h 1x h 2 ) = ζ p1n 2 for all x, y ∈ Z, if and only if i ord p (1 + p s ) p − 1 = s + i ≥ n,
i.e. if and only if i ≥ n − s.
123
D. Delbourgo, Q. Chao
" √ p s+r 1 + ps ; let λ I V,± := 1 + p s ± p s+r d be the two Case (IV). Here I2 + M = p s+r d 1 + p s distinct eigenvalues of I2 + M, so that ! ! " " 1 1 λ I V,+ 0 √ √ I2 + M = PI V D I V PI−1 withD = = . and P IV IV V 0 λ I V,− d − d !
pi
Since (I2 + M) p = PI V D I V PI−1 V , one readily computes i
γ
− pi
i y (h 1x h 2 )γ p
i pi λ IpV,+ +λ I V,− 2
= h1
i pi λ IpV,+ −λ I V,− √ 2 d
x+
y
i pi λ IpV,+ −λ I V,− √
× h2
i pi λ IpV,+ +λ I V,−
d x+
2
2
y
. (4)
pi
To study both
pi
λ I V,+ +λ I V,− 2
pi
and
pi
λ I V,+ −λ I V,− , 2
√ p p λ I V,± = 1 + p s (1 ± pr d) = 1+
note that
! " p ! " √ √ p p p s (1 ± pr d) + p s j (1 ± pr d) j 1 j j=2
√ √ and (1 ± pr d) j = 1 ± j pr d + O p 2r +δ p where δ p = ord p (d), hence p ! " p ! " √ √ p p p λ I V,± = 1 + p s+1 ± p s+r +1 d + p s j ± pr d j ps j j j j=2 j=2 2s+2r +1+δ p +O p √ = 1 + p s+1 ± p s+r +1 d + (1 + p s ) p − 1 − p s+1 ± O p 2s+r +1+δ p /2 + O p 2s+2r +1+δ p .
λ I V,+ +λ I V,− will equal 1+ p s+1 + (1+ p s ) p −1− p s+1 + O p 2s+r +1+δ p /2 , 2 p p λ +λ accurately I V,+ 2 I V,− = 1 + p s+1 + O p 2s+1 ; applying an induction argument: p
p
It follows that or less
pi
pi
λ I V,+ + λ I V,− 2
= 1 + p s+i + O p 2s+i .
√ λ I V,+ −λ I V,− equals p s+r +1 d + O p 2s+r +1+δ p /2 , 2 O p 2s+r +1 ; applying induction again: p
On the other hand, the difference term p
and therefore
p
λ I V,+ −λ I V,− √ 2 d
= p s+r +1 + pi
(5)
p
pi
λ I V,+ − λ I V,− = p s+r +i + O p 2s+r +i . √ 2 d
(6)
e1 e2 × χ2,n , from Eq. (4) one obtains Recalling the chosen character χ = χ1,n i i i i e1 i i e2 y y y χ γ − p (h 1x h 2 )γ p = χ1,n γ − p (h 1x h 2 )γ p × χ2,n γ − p (h 1x h 2 )γ p ⎛
⎝ e1
= ζ pn
123
i pi λ IpV,+ +λ I V,− 2
+e2
i pi λ IpV,+ −λ I V,− √ 2
⎞
⎛
d ⎠x+⎝e1
i pi λ IpV,+ −λ I V,−
i pi λ IpV,+ +λ I V,−
2 d
2
√
+e2
⎞ ⎠y
.
K1 -congruences for three-dimensional
i i i y y As a corollary of our estimates in (5) and (6), γ p χ(h 1x h 2 ) = χ γ − p (h 1x h 2 )γ p equals y
e x+e2 y
χ(h 1x h 2 ) = ζ p1n
for all x, y ∈ Z, if and only if
e1 p s+i + e2 p s+r +i d ≡ 0 (mod p n ) and e1 p s+r +i + e2 p s+i ≡ 0 (mod p n ), i.e. if and only if i ≥ n − s − min ord p (e1 + pr de2 ) , ord p ( pr e1 + e2 ) = n − s. ! " √ s+r 1 ps Case (V). Here I2 + M = ; let λV,± := 1 + p 2 ± p s V with p s d 1 + p s+r V = d + p 2r /4 denote the eigenvalues of I2 + M. Indeed for all i ≥ 0, one may write pi
λV,+ 0 pi (I2 + M) = PV PV−1 pi 0 λV,− !
" 1 1 √ r where PV = pr √ , and its inverse PV−1 = p + − V V 2 2 Using this decomposition, we next deduce λ pi
pi pi pi −λV,− r V,+ +λV,− λV,+ √ − × p2 2 2 V
γ − p (h 1x h 2 )γ p = h 1 λ pi i
y
i
× h2
pi −λ V,+ √ V,− 2 V
1 2
1− 1+
λ pi
pi −λ V,+ √ V,− 2 V
x+
λ pi
d x+
r √p 2 V r √p 2 V
√1 V 1 − √ V
.
pi pi pi −λV,− r V,+ +λV,− λV,+ √ + × p2 2 2 V
y
y
.
(7)
Now from the binomial theorem, p
λV,± = 1 +
p ! " pr # j # p s+r +1 p ps j ± p s+1 V + ± V . j 2 2 j=2
√ • If ord p ( V ) ≥ r then
where δ p = ord p (V ), hence
pr 2
±
√ V
j
=
pr 2
j
± j
pr 2
j−1√
V + O pr ( j−2)+δ p
! " ! r " j−1# p ! " # j pr j p p sj ± V = p p ±j V j j 2 2 2 j=2 j=2
+ O p 2s+1+δ p !
! "p ! " " p−1 # pr +s+1 pr +s pr +s s+1 − 1+ V × 1+ −1 = 1+ ±p 2 2 2
+ O p 2s+1+δ p .
p ! " p
sj
pr
p p s+r+1 λ +λ It follows that V,+ 2 V,− = 1 + p 2 + O p 2s+2r +1 and 2s+r +1 upon using the condition δ p ≥ 2r , so by induction: O p
pi
pi
λV,+ + λV,− 2
pi
p
p
λV,+ −λV,− √ 2 V
= p s+1 +
pi
λV,+ − λV,− p s+r +i = p s+i + O p 2s+r +i . =1+ + O p 2s+2r +i and √ 2 2 V (8)
123
D. Delbourgo, Q. Chao
• Alternatively, if r ≥ ord p
√
V then
# j
j pr # j−1 ± V ± V + + O p δ p ( j−2)/2+2r 2 2 and arguing in an identical fashion to before, one deduces that pr
pi
#
±
V
j
pi
λV,+ + λV,− 2
=
pi
=1+
pi
λV,+ − λV,−
p s+r +i = p s+i + O p 2s+δ p /2+i . + O p 2s+δ p +i and √ 2 2 V (9)
e1 e2 × χ2,n , this time Eq. (7) implies Again as χ = χ1,n ⎛ ⎝
e1 − pi x y pi = ζ pn χ γ (h 1 h 2 )γ
λ pi
⎛
⎝e1
× ζ pn
pi pi pi −λV,− r V,+ +λV,− λV,+ √ − × p2 2 2 V
λ pi
pi −λ V,+ √ V,− 2 V
+e2
+e2 d
λ pi
pi −λ V,+ √ V,− 2 V
λ pi
pi pi pi −λ r V,+ +λV,− λV,+ √ V,− × p2 + 2 2 V
⎞
⎠x
⎞
⎠y
.
Exploiting our eigenvalue estimates in Eqs. (8) and (9) appropriately, it follows that i i y y e x+e y χ γ − p (h 1x h 2 )γ p equals χ(h 1x h 2 ) = ζ p1n 2 for all x, y ∈ Z, if and only if e2 d × p s+i ≡ 0 ( mod p n ) and e1 × p s+i + e2 × p s+i+r ≡ 0 ( mod p n ); the latter holds precisely when s + i ≥ n − ord p (e2 d) and s + i ≥ n − ord p (e1 + e2 pr ). " ! √ 1 ps ; let λVI,± := 1 ± p s pr t be its eigenvalues Case (VI). Here I2 + M = s+r p t 1 (note that t = 1 in (a) of the Classification Theorem, and t ∈ Z× p is not a square in (b)). Then " " ! ! λVI,+ 0 1 1 pi −1 pi √ √ . and PVI = (I2 + M) = PVI DVI PVI with DVI = 0 λVI,− pr t − pr t A straightforward calculation shows γ − p (h 1x h 2 )γ p λ pi +λ pi i
i
y
VI,+
= h1
VI,−
2
λ pi
x+
pi −λ VI,+ √ rVI,− 2 p t
y
√
× h2
pr t
λ pi
pi VI,+ −λVI,− 2
√ p pr t + . . . and clearly λV,± = 1 ± p s+1 pr t + p 2s+1 p−1 2 2s+r +1 O p . Using a now familiar mathematical induction, pi
pi
λVI,+ + λVI,−
pi
λ pi
pi VI,+ +λVI,− 2
x+
=
y
(10)
1 ± p s+1
√
pi
pr t +
λVI,+ − λVI,− = 1+ O p = p s+i + O p 2s+r/2+i . (11) and √ r 2 2 p t i i y e1 e2 If the character χ = χ1,n × χ2,n , by Eq. (10) the value χ γ − p (h 1x h 2 )γ p equals ⎛ ⎝e1
λ pi
ζ pn
123
pi VI,+ +λVI,− 2
+e2
√
2s+r +i
pr t
λ pi
pi VI,+ −λVI,− 2
⎞
⎛
⎠x+⎝e1
λ pi
pi −λ VI,+ √ rVI,− 2 p t
+e2
λ pi
pi VI,+ +λVI,− 2
⎞
⎠y
.
K1 -congruences for three-dimensional
i i y y Plugging Eq. (11) into the above, one can then deduce χ γ − p (h 1x h 2 )γ p = χ h 1x h 2 for all x, y ∈ Z, if and only if both # 2 pr t ≡ 0 ( mod p n ) and e1 × p s+i ≡ 0 ( mod p n ), e2 × p s+i × which is itself equivalent to ensuring that s + i ≥ n − ord p (e2 pr t) = n − r − ord p (e2 ) and s + i ≥ n − ord p (e1 ).
2.2 A “coarse but clean” system of subgroups The theory in [7,16,17,23] operates best in the setting of one-dimensional Lie!groups. " Throughout we choose an integer n, and work with the p-adic group G∞,n := Hp∞n . H∞
In later sections we will allow n to vary, but for the time being n is fixed. Lemma 6 If Z (G) denotes the centre of a group G, then ⎧ pn−s ⎪ p n−s × H1,∞ ×H2,∞ ⎪ ⎪ in Case (I I ) n p ⎪ ⎪ H∞ ⎪ ⎪ ⎪ pn−s ⎪ ⎪ ⎪ pn−s × H∞ n in Cases (I I I )and(I V ) ⎪ p ⎨ H∞ Z (G∞,n ) = pn−s−ord p (d) pn−s ⎪ ×H2,∞ ⎪ ⎪ pn−s × H1,∞ in Case (V ) ⎪ pn ⎪ ⎪ H∞ ⎪ ⎪ ⎪ ⎪ pn−s−r−ord p (t) pn−s ⎪ × H2,∞ ⎪ ⎩ pn−s × H1,∞ in Case (V I ). pn H∞
H1,∞ in Case (I I )
In particular, Z (G∞ ) ∼ = limn Z (G∞,n ) = ← − {1}
otherwise.
Proof We first note from the semi-direct product structure on G∞,n that ! " ! " % $ x x y
≡ mod p n Z2p h 1x h 2 (I2 + M) y y H∞ Z (G∞,n ) = Stab . × pn pn H∞
H∞
One then computes the right-hand side on a case-by-case basis, using the form of the matrix M listed in Eq. (3) (see [5, Chap. 4] for the full details of each calculation). Bearing in mind Kakde’s subgroups should always contain the centre of G∞,n , we define
m
Um,n := p
H∞
where the integer m ∈ {0, . . . , n − s},
pn
H∞
n−s
so: (i) Z (G∞,n ) ⊂ Um,n , and (ii) p ⊂ Stab (χ) for any χ : H∞ μ pm by Proposition 5. It follows that such χ extend to Um,n if m ∈ {mχ , . . . , n − s}, and will thus factor through ab Um,n
Um,n = [Um,n , Um,n ]
m
= &
y
h 1x h 2
pn
p H∞ /H∞ ' . pn m mod H∞ , γ p x, y ∈ Z
123
D. Delbourgo, Q. Chao ab in each case, we may calculate the number Therefore, by determining the nature of Um,n G
(χ) with ψ : → C× of finite order. (Remember of irreducible representations ψ ⊗ IndU∞,n m,n that every irreducible Artin representation ρ on G∞ is of this form for suitable m, n, χ, ψ.) Proposition 7 For each pair m, n ∈ Z with 0 ≤ m ≤ n − s, ⎧ pm H H2,∞ × p1,∞ in Case (I I ) ⎪ s+m × pn ⎪ ⎪ H2,∞ H ⎪ 1,∞ ⎪ ⎨ Um,s+m in Cases (I I I ) and (I V ) ab ∼ Um,n = m Z Z p ⎪ × min{n,s+m+ ord p (d)} × ps+m Z in Case (V ) ⎪ ⎪ p Z ⎪ m ⎪ Z Z ⎩ p × × ps+m in Case (V I ); min{n,s+m+r+ ord p (t)} Z Z
p
in fact, the first two lines are actual equalities, not just isomorphisms. Proof We proceed by working through the different cases (II)–(VI) in numerical order. m
y y s+m Case (II). Here one simply exploits the commutator relation h 1x h 2 , γ p = (h 1 ) p . m x y pm
s p m y Case (III). Here we use h 1 h 2 , γ = (h 1x h 2 )(1+ p ) −1 and ord p (1+ p s ) p −1 = s+m. Case (IV). Recall from Eq. (4) that
γ
− pm
y (h 1x h 2 )γ
pm
m pm λ IpV,+ +λ I V,− 2
m pm λ IpV,+ −λ I V,− √ 2 d
x+
m pm λ IpV,+ −λ I V,− √
y
m pm λ IpV,+ +λ I V,−
d x+
2 = h1 × h2 s+m p +... p s+r+m d+... x p s+r+m +... p s+m +... y y = h1 × h2 × h1 × h2 × h 1x h 2 2
upon using the estimates in (5) and (6); consequently
H∞
m m [h 1 , γ p ], [h 2 , γ p ]
∼ =
Zp ⊕ Zp Z p · ( p s+m + . . . , p s+r +m d + . . . ), ( p s+r +m + . . . , p s+m + . . . ) Um,n
ab = which means Um,n
m
m ∼ = p ×
m
[h 1 ,γ p ],[h 2 ,γ p ]
Zp p s+m Z p
Zp . p s+m Z p
×
Case (V). This time Eq. (7) combined with the estimates (8) and (9) yields λ pm γ − p (h 1x h 2 )γ p m
! =
y
m
= h1
pm pm pm −λV,− r V,+ +λV,− λV,+ √ − × p2 2 2 V
λ pm
× h2 ps+r+m 2
h1
−p
ab = so that Um,n
s+r+m 2
+...
pm −λ V,+ √ V,− 2 V
m
pm −λ V,+ √ V,− 2 V
y
pm pm pm −λ r V,+ +λV,− λV,+ √ V,− × p2 + 2 2 V
"x
! ×
m ∼ = p ×
[h 1 ,γ p ],[h 2 ,γ p ]
λ pm
x+
λ pm
d x+
p s+m d+... × h2
Um,n m
p s+m +... h1
ps+r+m 2
× h2
Zp p n Z p ∪ p s+m dZ p
×
y +p
s+r+m 2
Zp . p s+m Z p
Case (VI). Lastly, Eq. (10) in tandem with the estimates in (11) implies
123
+...
"y y
× h 1x h 2
y
K1 -congruences for three-dimensional
λ pm γ
− pm
m y (h 1x h 2 )γ p
ab = hence Um,n
pm VI,+ +λVI,− 2
λ pm
x+
pm −λ VI,+ √ rVI,− 2 p t
y
√
pr t
λ pm
pm VI,+ −λVI,− 2
λ pm
pm VI,+ +λVI,− 2
x+
y
= h1 × h2 y s+m p s+m+r t+... x p +... y 0+... = h1 × h2 × h1 × h 0+... × h 1x h 2 , 2 Um,n m
m
m ∼ = p ×
[h 1 ,γ p ],[h 2 ,γ p ]
Zp p n Z p ∪ p s+m+r tZ p
Zp . p s+m Z p
×
(m,n)
m
ab has the form p × H We remark in Cases (II-VI), each Um,n ∞ pn
(m,n)
where H∞
from quotienting H∞ /H∞ = h 1 , h 2 with the subgroup generated by is obtained m m [h 1 , γ p ], [h 2 , γ p ] . (m,n) m (m,n) ” denote the orbits under the action of p in H∞ . In Definition 8 Let “orb H∞ (m,n) −i i (m,n) consists of the set γ hγ i ∈ Z ; we particular, if h ∈ H∞ then h ∈ orb H∞ shall sometimes abuse notation, and write h in place of h .
2.3 Maps between the abelianizations of Um,n We now outline the various mappings that appear in the description of and in [7,17]. Rather than give their full definitions, we specialise them to the specific three-dimensional situation we are considering. The conditions (A1)–(A3) and (M1)–(M4) in the exposition [7, p 79–123] degenerate into some fairly simple rules, which can be expressed in terms of an explicit basis for the image N (U ) of Kakde’s map “σU ”. In subsequent sections we will then study how these expressions ab are evaluated at a system of characters transform, once the completed group algebras Um,n χ on H∞ . The mapping σm : Note that the normaliser of each subgroup U = Um,n ⊂ G∞,n is the N (U ) whole of G∞,n , so the Z p -linear map labelled σU in [7, p85] becomes p −1 ab ab : Um,n −→ Um,n where f → γ −i f γ i . m
G∞,n σUm,n
i=0
ab If we use the shorthand σm for this linear mapping, clearly σm ( f ) ∈ H 0 , Um,n m corresponds to the sum over the orbits of f under the action of the finite group / p .
ab
y (m,n) Definition 9 For any h = h 1x h 2 mod Um,n , Um,n , one defines A ∈ Z p Um,n by h
A
(m,n) h
p m −1
:=
x
!
y
h 1i h 2i where
i=0
xi yi
"
In fact, we could alternatively have defined A
≡ (I2 + M)i (m,n) h
! " x mod p n . y
to be the summation
( pm −1 i=0
γ −i hγ i
which coincides, of course, with σm (h); we will see that these form a basis for Im(σm ). (m,n) h
(m,n)
depends only on the -orbit of h inside H∞ ; m
(m,n) ’s, in other words (ii) The image of σm is freely generated over Z p p by the A h m
y (m,n) Im σm ∼ h = h 1x h 2 mod Um,n , Um,n ; = Z p p ⊗Z p Z p Ah
Proposition 10 (i) Each element A
123
D. Delbourgo, Q. Chao
(n) (iii) If rσm := rankZ p [[ pm ]] Im(σm ) , then
rσ(n) m
⎧ n+s−1 p × (mp + p − m) ⎪ ⎪ ⎪ ⎨ p 2s−1 × ( p m+1 + p m − 1) = ⎪ p min{n−m,s+ ord p (d)}+s−1 × ( p m+1 + p m − 1) ⎪ ⎪ ⎩ min{n−m,s+r + ord p (t)}+s−1 × ( p m+1 + p m − 1) p
in Case (I I ) in Cases (III) and (IV) in Case (V ) in Case (VI). (m,n)
m
ab = p × H Proof Statement (i) is self-evident. To establish (ii), first note that Um,n ∞ m (m,n) where H∞ is the previous quotient of H∞ equipped with the action of the group p ;
(m,n) y part (ii) now follows because H∞ is generated by h 1x h 2 mod Um,n , Um,n for x, y ∈ Z. (m,n) To prove (iii) we just need to count the number of distinct A ’s, which coincides with h m (m,n) p the total number of / -orbits inside H∞ . In fact by Burnside’s lemma,
# -orbits
(m,n) inH∞
pm (m,n) p m −1 × # h ∈ H∞ γ − j hγ j = h . = # / j=1 (m,n)
From Proposition 7, in each case ∈ {II,III,IV,V,VI} one knows H∞ (m)
(m)
(m)
∼ = (m)
Z
p
(m) N,1
Z
Z
× p
(m) N,2
Z
where N,1 , N,2 ∈ N satisfy m + s ≤ N,1 ≤ n and m + s ≤ N,2 ≤ n in all five scenarios. • Assuming that = II, one discovers pm
N (m) + ord ( j)−m (m) (m,n) p # -orbits inH∞ = p −m × p ,1 × p N,2 + ord p ( j)−m = p
j=1
(m)
(m)
N,1 −m + N,2 −m −1
× p m+1 + p m − 1 . (m,n)
• Alternatively, if = II then γ acts trivially on the first direct factor in H∞
, whence
pm
N (m) (m) (m,n) # -orbits inH∞ p II,1 × p N II,2 + ord p ( j)−m = p −m × = p
j=1 (m) N II,1 −m
(m)
+ N II,2 −m −1 (m)
× (m + 1) p m+1 − mp m . (m)
The result follows upon plugging in values of N,1 and N,2 listed in Proposition 7.
G
∞,n Corollary 11 The number of irreducible representations of the form Ind Stab n (χ) (χ )H∞ / p
(m,n)
where χ factors through H∞
(m−1,n)
but not through H∞
(n)
(n)
is given by rσm − rσm−1 .
Proof Note that any two characters χ, χ as above induce the same G∞,n -representation, if (m,n) and only if χ belongs to the -orbit of χ inside Hom H∞ , C× ; since the latter group is (m,n)
(non-canonically) isomorphic to H∞ , its -orbits are in one-to-one correspondence with (m,n) the finite set orb H∞ . It follows immediately that (m,n) (m−1,n) (m,n) − # orb H∞ , “the no. of Ind(χ) s primitive onH∞
= # orb H∞
123
K1 -congruences for three-dimensional
(m,n) m (n) (n) ∈ orb H(m,n) . which equals rσm − rσm−1 because Im(σm ) = Z p [[ p ]] · A ∞ h h The transfer map Verm,m . Consider the subgroups Um,n ⊂ Um ,n of G∞,n with m > U
m .
The transfer homomorphism (Verlagerung) VerUmm,n,n relative to these subgroups maps ab by sending Umab ,n −→ Um,n
cg,τ Um,n , Um,n g Um ,n , Um ,n → τ ∈R
where R is a fixed set of left coset representatives for Um ,n Um,n , and gτ = r g cg,τ with cg,τ ∈ Um,n and r g ∈ R. ab Henceforth one writes Verm ,m : Umab ,n → Um,n for the Z p -linear and continuous extension of the transfer map to the completed group algebras. m
m
y
Lemma 12 Suppose g ∈ Umab ,n , and let gˆ = (γ p ) j · (h 1x h 2 ) ∈ p H∞ be any lift. Then
m
y
Verm ,m (g) ≡ (γ p ) j · h 1x h 2 mod Um,n , Um,n
where (x , y ) = p m−m x, p m−m y in Case (II), and in the same notation as the proof of Proposition 5: ⎛ pm ⎞ λ,+ −1 0 ! " ! "
⎜ pm ⎟ x ⎜ λ,+ −1 ⎟ −1 x P = P in Case(), with ∈ {III,IV,V,VI}. ⎟ ⎜ pm y y
λ,− −1 ⎠ ⎝ 0
pm λ,− −1
m m Proof Since Um ,n Um,n ∼ = p p , its coset representatives are r0 , r1 , . . . , r pm−m −1 where ri = γ p in which case
m i
. One can represent gˆ in the form γ p
gˆ ri = γ p where
x pm i y pm i
m
m
y
(h 1x h 2 )γ p i = γ p x m y m
m = γ p ( j+i) · h 1 p i h 2 p i j
m
m ( j+i)
y
· (h 1x h 2 ) for some choice of j ∈ Z p ,
j
γ −p
m i
y
(h 1x h 2 )γ p
m i
! " pm i x
. In fact, if ι : Z p → {0, 1, . . . , p m−m − 1} so that = I2 + M y
ι(z) ≡ z mod p m−m , then γ p
m ( j+i)
gˆ ri = rι( j+i) γ
= rι( j+i) · γ p
m ( j+i−ι( j+i))
p m ( j+i−ι( j+i))
x
pm i
y
m i
· h1
; consequently m
h2 p i . y
By definition, the transfer is congruent to
Verm ,m (g) ≡
p m−m −1
γp
m ( j+i−ι( j+i))
x
· h1 p
m i
h2 p
mod Um,n , Um,n
i=0
m
p ( j+i−ι( j+i)) ∈ p , hence γ p and as j + i ≡ ι( j + i) mod p m−m clearly
γ xi yi and h 1 h 2 commute modulo Um,n , Um,n . It follows that
m
y
Verm ,m (g) ≡ γ p c · h 1x h 2 mod Um,n , Um,n m
m ( j+i−ι( j+i))
123
D. Delbourgo, Q. Chao
where c =
( pm−m −1 i=0
j + i − ι( j + i), and the vector
(
! " x pm i x ( = = y
y pm i
p m−m −1 i=0
! " pm i x . I2 + M y
To calculate the term c, without loss of generality assume j ∈ Z, which implies
c =
p m−m −1
j + i − ι( j + i) = p
m−m
×
i=0
p m−m −1 + i=0
j +i ,
. p m−m
The right-hand sum then yields
p m−m −1 + i=0
+ j , j +i , m−m
= p +
p m−m p m−m = p
m−m
+
,
j p m−m
= p m−m
+
i=0
ι( j) + i ,
p m−m
p m−m −ι( j)−1
+
p m−m −1
0 +
1
i=0
,
j
p m−m
p m−m −1 +
i= p m−m −ι( j)
+ ι( j) = j
and as an immediate consequence, c = p m−m × j so that γ p To compute x and y , in Case (II) we find that
m c
=γp
m
j
as required.
m−m −1 m−m p s+m × p p 2 I2 + M = = .
0 p m−m i=0 i=0 ⎛ ⎞
pm i pm i λ 0 −1 = P ⎝ ,+ In all other cases ∈ {III,IV,V,VI} one has I2 + M
⎠ P , which pm i 0 λ,− means ⎛ ⎞
! " ( pm−m −1 pm i p m−m −1 pm i x λ 0 ,+ ⎠ −1 I2 + M = P ⎝ i=0 ( pm−m −1 pm i P . y λ 0 i=0 ,− i=0
p m−m −1
pm i
p m−m −1 !
1 ps × i pm 0 1
"
Note that PIII = I2 because I2 + M is already diagonalised. The result follows upon summing pm ( pm−m −1 pm i λ −1 λ,± equals p,± . up the relevant geometric progression, i.e. i=0 m
λ,± −1
mapping to Verm ,m . The shift πm,m . For integers m > m , we now look for a reverse !x " m m y x p p The commutator [h 1 h 2 , γ ] corresponds to (I2 + M) − I2 as a vector in Z2p ; y m m
d however X p −1 = (X p −1)× m d=m +1 φ p d (X ) where φ p d denotes the p -th cyclotomic polynomial, therefore !
" ! " m x m
y
m
x y I φ + M = [h 1x h 2 , γ p ] = [h 1x h 2 , γ p ] with . d 2 p y
y
d=m +1
123
K1 -congruences for three-dimensional
pn As a consequence, we have the containments Um,n , Um,n ⊂ Um ,n , Um ,n ⊂ H∞ H∞ . The natural inclusion Um,n → Um ,n then yields the composition ab = πm,m : Um,n
Um,n Um,n , Um,n
→
Um ,n Um,n , Um,n
proj
Um ,n
Um ,n , Um ,n
= Umab ,n .
(m,n) → Moreover this shift homomorphism induces a map (πm,m )∗ : orb H∞ (m ,n) −i i , sending each orbit h = γ hγ i ∈ Z to the direct image π (h) . orb H∞ m,m Recall from Proposition 10(ii) that a typical element of Im(σm ) has the form m f (γ p − 1) · A(m,n) = f · A(m,n) say, (m,n)
∈ orb (H∞
)
( pm −1 (m,n) where each f (X ) ∈ Z p [[X ]] and A := i=0 γ −i hγ i for any h ∈ . ( (
(m,n) (m ,n) = p m−m × f · A(π )∗ ( ) . Lemma 13 If m > m , then πm,m
f · A m,m
(m,n) Proof If h ∈ with ∈ orb H∞ , then within the algebra Umab ,n one has ⎛ ⎞ ⎛ m−m
⎞ m −1 m −1 p p −1 p
m m m πm,m ⎝ f · γ −i hγ i ⎠ = f (γ p −1) · πm,m ⎝ γ − p i1 −i2 hγ p i1 +i2 ⎠ i 1 =0
i=0 m
= f (γ p − 1) · m
p m−m −1
p m −1
i 1 =0
i 2 =0
i 2 =0
γ −i2 πm,m (h)γ i2
m
since γ − p πm,m (h)γ p = πm,m (h) inside Umab ,n , which gives the result.
The norm and trace homomorphisms. We now introduce two final maps that occur in the definition of both of Kakde’s groups and . Firstly, if G is a group and Conj(G) denotes it set of conjugacy classes, then Conj(G) ∼ = (G)/[ (G), (G)] as an isomorphism of Z p -modules [7, Schneider-Venjakob, Lemma 2.1] For an integer pair m, m with m ≥ m :
relative to the • the norm mapping K 1 Umab ,n −→ K 1 Um,n Um ,n , Um ,n Um,n [Um ,n ,Um ,n ]
relative to
Um,n [Um ,n ,Um ,n ]
⊂
Um ,n [Um ,n ,Um ,n ]
= Umab ,n is abbreviated by Nm ,m ; and
• similarly, the additive trace map Conj Umab ,n −→ Conj Um,n Um ,n , Um ,n subgroup
⊂
Um ,n [Um ,n ,Um ,n ]
= Umab ,n is denoted by Trm ,m .
The following lemma describes the effect of the second of these maps on the image of m σm . Let char pm : () → ( p ) denote the Z p -linear and continuous extension of the map which sends γ i → γ i if p m divides i, and sends γ i → 0 if p m does not divide i.
( m
(m ,n) Lemma 14 For a typical element am = f (γ p − 1) · A ∈ Im σm 1/ p , ! " m
Um,n
(m ,n) char pm f γ p − 1 · A
∈ Trm ,m am = p m−m × [Um ,n , Um ,n ]
(m ,n) . where the sum is taken over all ∈ orb H∞
123
D. Delbourgo, Q. Chao
m Proof From [7, Rk iii], one knows Trm ,m γ p j h = so that for any h ∈ :
Trm ,m γ
pm j
(m ,n) · A
( pm −1
p m−m × 0
=
i=0
γp
p m−m × γ p
m j
if γ
0 m j
h if γ p
m j
pm j
∈ p
m m
∈ / p ,
m
m · γ −i hγ i if γ p j ∈ p otherwise.
The stated formula then follows by linearity and continuity.
3 The additive calculations ab
given in [17]. We begin by recalling Kakde’s definition of the subset ⊂ m Q p Um,n For a fixed n ≥ s, the Z p -module consists of sequences am satisfying the conditions: (A1) Trm ,m am = πm,m am for anym > m ; (A2) am = gam g −1 at everyg ∈ G∞,n ; (A3) am ∈ Im(σm ) for eachm ∈ {0, . . . , n − s}. In fact, the general definition of involves more than just this system of sub-quotients. However the “coarse but clean” choice of subgroups we made is sufficient for our purposes, as every irreducible representation of G∞,n is a finite twist of a representation obtained from inducing down a character χ on Um,n , for an appropriate choice of m and χ. (m,n)
3.1 The image of under the characters on H∞
The main task is to see how transforms if we evaluate its constituent elements at a system of pn characters χ = {χ} on H∞ /H∞ . In particular, we want to translate the conditions (A1)–(A3) involving the am ’s into equivalent conditions involving aχ = χ(amχ ) instead, and thereby complete the middle square in the diagram
K1 Zp [[G∞,n ]]
Θ∞,n
−→ Ω ⏐
Evχ
“twisted log”
−→
⏐
χ
χ Ω
Ψ
??
⎛
⎞
→ Q⊗⎝
ab ⎠ Zp Um,n
0≤m≤n−s
⏐
χ
χ(Ψ) → Q ⊗
χ
Oχ StabΓ (χ)
.
m,χ
C p Stab (χ) using p-adic congruences.
Theorem 15 A collection of elements aχ ∈ OC p Stab (χ) arises from a sequence (am ) ∈ ab
(m,n) ∩ 0≤m≤n−s Z p Um,n , if and only if for each m ≥ 0 and ∈ orb H∞ : The following key result describes χ() ⊂
χ
(C1) the compatibility χ(am ) = Tr Stab (χ )/ pm (aχ ) holds if m ∈ {mχ , . . . , n − s}, (C2) the equality aχ = aχ holds at each character χ ∈ ∗ χ, pm
( ∗ m (C3) , and χ ∈Rm,n Tr Stab (χ )/ p (aχ ) · Tr Indχ ( ) ∈ Z p (m,n) ( ∗ ord p (# H∞ )+m− ord p (# ) m (C4) χ ∈Rm,n Tr Stab (χ )/ p (aχ ) · Tr Indχ ( ) ≡ 0 mod p (m,n) where Rm,n denotes a set of representatives for the -orbits inside Hom H∞ , C× .
123
K1 -congruences for three-dimensional (m,n)
To calculate #H∞ in property (C4) above, one just applies Proposition 7. On the other hand, to calculate # we use the orbit-stabilizer theorem, so that for any h ∈ one obtains
m # = / p : Stab/ pm (h) = : Stab (h) . Also by property (C2), an element aχ depends only on the representative for χ in Rm,n , hence the last two summations in the above theorem are independent of any choices. Proof We begin with the ‘only if’ part of the argument. Suppose we are given an arbitrary ab
(m) , and let us put aχ := χ (am ) for any character χ : H∞ → μ pn element am ∈ Z p Um,n m (m) above completely). (note that if Stab (χ) = p , then we
will drop the superscript ab Assuming that (am ) ∈ ∩ m Z p Um,n , we claim the following statements hold: (m) (m) (a) there are equalities aχ = aχ for any χ ∈ ∗ χ, where ∗ χ := g ∗ χ g ∈ ; ( (m) (m,n) (b) we can express am = ∈ orb (H(m,n) ) C · A , where for any h ∈ one has
#
(m) = C
(m,n)
p m · # H∞ ! "
# (c) − ord p (m,n) p m ·#H∞ ∗ (d) Tr Indχ ( ) = (m)
(e) one has aχ
×
χ ∈Rm,n
∞
⎞ m −1 p m #( ∗ χ) aχ(m) · ⎝ · χ −1 γ −i hγ i ⎠ ∈ p ; m p ⎛
i=0
(m,n) = ord p #H∞ + m − ord p (# ) ≥ 0; m ( p −1 −1 −i #(∗χ ) γ hγ i ; i=0 χ pm ·
= Tr Stab (χ )/ pm (aχ ) for each m ≥ mχ , i.e. (C1) is true.
Deferring their proof temporarily, let us first understand why they yield the three assertions in our theorem. Clearly statement (C2) is implied by (a) with m = ord p [ : Stab (χ)]. Moreover both (C3) and (C4) will now follow upon combining (b), (c), (d) and (e) together, (m) and then observing that the p-integrality of the C ’s is equivalent to each sum ⎞ ⎛ m −1 p #( ∗ χ) aχ(m) · ⎝ · χ −1 γ −i hγ i ⎠ pm i=0 χ ∈Rm,n = Tr Stab (χ )/ pm (aχ ) · Tr Indχ ∗ ( ) χ ∈Rm,n
(m,n) (m,n) m
m
p m ·#H∞ belonging to the lattice · Z p p = p ord p (#H∞ )+m− ord p (# ) · Z p p . # We are left to prove these five assertions. Part (a) is a consequence of property (A2). To m ( (m) (m) prove statement (b), let us write am = h∈H(m,n) ch · h where each ch ∈ p . Since ∞
the characteristic function of h can be decomposed into a sum over the characters of the (m,n) abelian group H∞ , one can express each coefficient above as c
(m) h
=
1 (m,n) # H∞
×
χ −1 (h) · aχ(m) .
(m,n)
χ :H∞ →μ pn m
Using property (A3) and Proposition 10, we know that am is a ( p )-linear combination (m,n) (m) of A ’s, which indicates ch is constant-valued for all h inside a prescribed orbit . If (m)
we denote this common value as ‘c ’, then
123
D. Delbourgo, Q. Chao
am =
(m,n)
∈ orb (H∞
(m) c ·h =
) h∈
(m) c ·
h =
h∈
(m) c ·
(m)
# · A(m,n) . pm
(m)
(m,n)
of A . N.B. In this situation, the term c · # p m corresponds to the coefficient C ( ( ( Now we can always break χ :H(m,n)→μ n into a double summation χ ∈Rm,n χ ∈∗χ . (m)
(m)
Furthermore, aχ = aχ (m) = c
1 (m,n) # H∞
∞
·
p
whenever χ ∈ ∗ χ from (a), hence for any h ∈ : χ −1 (h) · aχ(m) =
(m,n)
χ :H∞ →μ pn
1 (m,n) # H∞
·
χ ∈Rm,n
aχ(m)
−1 χ (h). χ ∈∗χ
Splicing together these last two equations, we therefore conclude ⎞ ⎛ −1 # ⎝ χ (h)⎠ · A(m,n) am = × aχ(m) · . (m,n) p m · # H∞
(m,n) χ ∈Rm,n χ ∈∗χ ∈ orb (H∞
)
−1 ) ( p m −1 −1 −i γ hγ i , · i=0 χ Lastly χ ∈∗χ χ (h) coincides with the scaled value #(∗χ pm which means (b) is also established. (m,n) To show part (c) is easy since the size of each orbit ∈ orb H∞ divides into p m . G ∞,n ∼ In order to establish (d) we define ρm := Ind m (χ), so that ρm = Ind(χ) ⊗ ψ (
ψ
p H∞ / p n
(m,n)
where the sum is over all characters ψ : Stab (χ)/ p → C× . Thus for h ∈ ⊂ H∞ , m
Stab (χ) :
pm
· Tr Indχ ∗ (h) =
Tr ρm∗ (h) =
m −1 p
χ −1 γ −i hγ i
i=0
(m,n) m and the orbit-stabilizer theorem for / p acting on Hom H∞ , μ pn then implies m
m
: p : p pm m
=
= . = Stab (χ) : p m : Stab (χ) / p : Stab/ pm (χ) # ∗χ The assertion (e) follows from property (A1): if we set m = mχ then Tr Stab (χ )/ pm (aχ ) = χ Trm ,m am
by (A1)
=
χ πm,m am = aχ(m) .
Finally, we must of course demonstrate the ‘if’ portion of the ‘if and only if’ statement. This amounts to showing the implication “(A1) and (A2) and (A3) ⇒ (C1) and (C2) and (C3) and (C4)” is in fact reversible, which is a tedious but relatively straightforward exercise involving Lemmas 13 and 14 – we refer the reader to [5] for further details.
3.2 A transfer-compatible basis for the set Rm,n (m,n)
Assume again that ∈ {II, III, IV, V, VI}. We can express H∞ (m,n)
H∞
123
∼ =
pn
H∞ /H∞
m m [h 1 , γ p ] , [h 2 , γ p ]
as the double quotient
K1 -congruences for three-dimensional pn
where h 1 and h 2 denote the image inside H∞ /H∞ of the subgroup generators h 1 , h 2 ∈ H∞ , as outlined in the Classification Theorem. m m (m,n) Clearly any character χ defined on H∞ must satisfy χ [h 1 , γ p ] = χ [h 2 , γ p ] = (m,n) ∼ (m) (m) Z Z 1. Also H × (m) where N , N ∈ N can be read off from Proposition = (m) ∞
p
N,1
7; one may then write
h1, γ p
m
Z
p
N,2
,1
Z
,2
(m)
N y˜ p ,1 = h 1x˜1 h 21
and
h2, γ p
m
(m)
N y˜ p ,2 = h 1x˜2 h 22
Z Z for integer pairs (x˜1 , y˜1 ) and (x˜2 , y˜2 ), neither of which is p-divisible in (m) × (m) . N,1 N,2 Z p Z x y pm p corresponds to the To precisely determine them, we note that the commutator h 1 h 2 , γ ! " m x vector (I2 + M) p − I2 inside Z p ⊕ Z p , whence y
(m) " ! p −N,1 pm 0 x˜1 x˜2 − I2 . (12) = I2 + M (m) y˜1 y˜2 0 p −N,2
(m,n) To construct a basis for Hom H∞ , C× , we therefore need a pair of characters χ˜ 1 x˜
(m)
y˜
and χ˜ 2 , sending h 1 j h 2 j to a primitive p N, j -th root of unity for each j ∈ {1, 2}. Recall the definition of the generating characters χ1,n , χ2,n : H∞ → μ pn from §1.2, namely √ √ y y and χ2,n h 1x h 2 = exp 2π −1 y/ p n . χ1,n h 1x h 2 = exp 2π −1 x/ p n (m,n)
As an illustration, in Case (II) we know H∞ one may set y y y χ˜ 1,N (m) h 1x h 2 := χ2,n h 1x h 2 = ζ pn
H1,∞
∼ =
ps+m
H1,∞
×
H2,∞ pn
H2,∞
from Proposition 7, thus
y y and χ˜ 2,N (m) h 1x h 2 := χ1,s+m h 1x h 2 = ζ pxs+m .
II,1
II,2
!" x y We will now abuse our notation, and employ χ as an abbreviation for χ(h 1x h 2 ). y (m,n)
Definition 16 For j ∈ {1, 2}, we define characters χ˜ j,N (m) : H∞ , j
• if ∈ {III, IV, V, VI }, then (m) !" x p N,1 χ˜ 1,N (m) := χ1,N (m) y ,1 ,1 0 and
!" 0 x χ˜ 2,N (m) := χ2,N (m) ,2 y ,2 0
0 0
0
(m)
p N,2
I2 + M
I2 + M
pm
pm
μ
− I2
− I2
p
(m) N, j
(13)
through:
−1 !x "
y
−1 !x " y
;
• if = II , one uses Eq. (13) instead to define χ˜ 1,N (m) and χ˜ 2,N (m) . II,1
II,2
In particular, from Eq. (12) we see that χ˜ 1,N (m) h 1x˜1 h 2 = χ1,N (m) h 11 h 02 = ζ N (m) and ,1 ,1 p ,1 y˜ χ˜ 2,N (m) h 1x˜2 h 22 = χ2,N (m) h 01 h 12 = ζ N (m) , which satisfies our stated requirement. The ,2
,2
p
y˜1
,2
123
D. Delbourgo, Q. Chao
main reason why we prefer using the character set χ˜ 1,N (m) , χ˜ 2,N (m) over the more naive ,1 ,2 choice χ1,N (m) , χ2,N (m) is motivated by the following compatibility result. ,1
,2
(m,n) Proposition 17 (a) The elements of Hom H∞ , C× are explicitly given by the set $ χ˜ e1
(m)
1,N,1
· χ˜ e2
(m)
2,N,2
% (m) (m) where e1 ∈ Z p N,1 Z and e2 ∈ Z p N,2 Z .
(b) If = II and m > m , then χ˜ 1,N (m) ◦ Verm ,m = χ˜
pm−m
(m ) 1,N,1
,1
and χ˜ 2,N (m) ◦ Verm ,m = χ˜
.
(m )
2,N,2
,2
(c) If ∈ {III, IV, V, VI } and m > m , then χ˜ j,N (m) ◦ Verm ,m = χ˜ , j
(m )
j,N, j
at each j ∈ {1, 2}.
m
m
p s+m
with Proof Let us first suppose = II. Here one has [h 1 , γ p ] = 1 and [h 2 , γ p ] = h 1 x y x y y (m) (m) N II,1 = n and N II,2 = s + m, whilst χ˜ 1,N (m) h 1 h 2 = ζ pn and χ˜ 2,N (m) h 1 h 2 = ζ pxs+m . II,1
II,1
(m,n)
Part (a) then follows as χ˜ 1,N (m) and χ˜ 2,N (m) are independent, while #H∞ = p n · p s+m . To II,1 II,1
p m−m show (b) one notes for j = 1, 2 that χ˜ j,N (m) ◦ Verm ,m (m ,n) = χ˜ (m) by Lemma 12, in H∞
II, j
j,N II, j
which case x y m−m y pm−m
x y m−m x pm−m
χ˜ 1,N (m) (h 1 h 2 ) p = ζ pn = ζ ps+m and χ˜ 2,N (m) (h 1 h 2 ) p = ζ pxs+m . II,1
II,2
Let us instead suppose ∈ {III,IV,V,VI}. Since I2 + M
pm
= P
deduce that
(m)
p N,1 0 0 0
I2 + M
pm
− I2
−1
! =
10 00
"
⎛ ⎜ P ⎜ ⎝
N
pm
λ,+ 0 pm 0 λ,−
0
P−1 , we
⎞
(m)
p ,1 pm λ,+ −1
0 (m) N,1
p pm λ,− −1
⎟ −1 ⎟P . ⎠
On the other hand, again from Lemma 12 the matrix corresponding to Verm ,m (m ,n) is H∞ ⎛ pm ⎞ λ,+ −1 0 ⎟ ⎜ pm
⎜ λ,+ −1 ⎟ −1 given by P ⎜ ⎟ P . An elementary calculation reveals the identities pm λ,− −1 ⎠ ⎝ 0
m p λ,− −1
123
K1 -congruences for three-dimensional
(m)
p N,1 0
0 0
!
1 0
=
= p
0 0
(m)
⎛
"
I2 + M ⎛
⎜ P ⎜ ⎝
(m )
pm
− I2
(m) N p ,1 m p λ,+ −1
0 (m) N p ,1 pm λ,− −1
0
(m )
p N,1 0
N,1 −N,1
−1
0 0
⎞ ⎛ pm λ,+ −1
pm ⎟⎜ λ,+ −1 ⎟⎜ ⎜ ⎠⎝ 0
I2 + M
pm
(m )
p
(m )
N,1
,1
(m)
0
0 0
pm λ,− −1
pm
λ,− −1
0 pm λ,− −1
pm λ,− −1
− I2
I2 + M
! " ⎟ ⎟ −1 x ⎟ P y ⎠
⎞
,1
(m)
0
⎜ pm
⎜ λ −1 · P ⎜ ,+ ⎝ 0
These matrix identities directly imply that χ˜ 1,N (m) ◦ Verm ,m p N,1 −N,1 χ1,N (m)
⎞
pm
λ,+ −1
! " ⎟ ⎟ −1 x ⎟ P y ⎠
−1 !x " y
.
! " x equals y
pm
− I2
−1 !x "
.
y
(m )
p N,1 −N,1 = χ Since χ1,N (m)
(m ) 1,N,1
,1
the above quantity is none other than χ˜
(m ) 1,N,1
which establishes that χ˜ 1,N (m) ◦ Verm ,m = χ˜
(m )
1,N,1
,1
! " x , y
.
The argument for the second composition χ˜ 2,N (m) ◦ Verm ,m follows identical lines. ,2
x y
(m ,n)
Lemma 18 (i) If h 1 h 2 ∈ H∞
and f (X ) ∈ Z p [[X ]], then
m
m (m,n)
(m ,n) Verm ,m f γ p − 1 · A x y = p −(m−m ) × f γ p − 1 · A x y
h1 h2
h1 h2
where x , y are as in Lemma 12. (ii) Using exactly the same notation,
(m ,n) χ˜ e1 (m ) · χ˜ e2 (m ) A x y = p −(m−m ) × χ˜ e1 1,N,1
2,N,2
(m) 1,N,1
h1 h2
unless =II, in which case one replaces χ˜ e1 on the left-hand side of this formula. Proof Let us start by establishing (i). If
(m ) 1,N,1
· χ˜ e2
(m ) 2,N,2
· χ˜ e2
(m) 2,N,2
A
instead with χ˜
(m,n)
x y
h1 h2
e1 p m−m
(m ) 1,N,1
· χ˜ e2
(m )
2,N,2
! " ! " i x xi = I2 + M for all i ≥ 0, then yi y
i=0
i=0
m −1 m −1 p p m
m
mj x y
xi yi (m ,n) p j p j p = ·A x y Verm ,m γ · h1 h2 = γ · h 1i h 2i Verm ,m γ
h1 h2
upon applying Lemma 12. Here in Case () with ∈ {III,IV,V,VI}, the vector
123
D. Delbourgo, Q. Chao
⎛
⎞
pm
λ,+ −1
! " ⎜ pm
xi ⎜ λ,+ −1 = P ⎜ yi
⎝ 0 ⎛ i ⎜λ,+ · ⎜ = P ⎜ ⎝
0 pm λ,− −1
pm
! " ⎟ ⎟ −1 xi ⎟ P yi ⎠
λ,− −1
⎞
pm λ,+ −1
pm λ,+ −1
0 λi,− ·
0
pm λ,− −1
pm λ,− −1
! " ! " ⎟ i x
⎟ −1 x = I2 + M ⎟ P y
y ⎠
m
m ( p m −1 m
x y
(m ,n) (m,n) so that Verm ,m γ p j · A x y equals γ p j · i=0 γ −i h 1 h 2 γ i = γ p j · p m −m A x y
h1 h2
h1 h2
(the same identity for the Verlagerung holds in Case (II) also). The result extends to the completed group algebra by linearity and continuity. Secondly to show part (ii) is true, we first set f (X ) = 1 and then evaluate the identity from (i) at the character χ˜ e1 (m) · χ˜ e2 (m) . We next use Proposition 17(b)-(c) to rewrite the 1,N,1
2,N,2
transformed left-hand side in terms of the powers of χ˜
(m )
1,N,1
and χ˜
(m )
2,N,2
.
4 The multiplicative calculations To complete the proof of the main theorem, our strategy is to establish the existence, commutativity and row-exactness of the diagram ab 1 → F× p × G∞,n → K1 Zp [[G∞,n ]] ⏐
LOG
−→ Zp Conj(G∞,n ) ⏐ Θ+ ∞,n
Θ∞,n
1 →
F× p
×
ab G∞,n
→
ab 1 → F× p × G∞,n →
Φ ⏐
L
−→
Ψ ⏐
χ
χ Φ
−→
χ Ψ →
m,χ
ab → G∞,n → 1
χ
Lχ
→ OCp StabΓ (χ)
ab → G∞,n → 1
×
OCp StabΓ (χ) m,χ
⊗ Zp Q p .
(14)
The top two lines of this diagram are precisely those occurring in [7, p80]. The vertical arrows ab ∼ , labelled as “χ ” denote evaluation at a system of representatives Rm,n , and as G∞,n = × ab the whole ensemble χ therefore restricts to being the identity map on F p × G∞,n . At this preliminary stage, we make no attempt to explain the maps LOG, L and Lχ . ab
will consist of elements satisfying From Sect. 3, the module ⊂ m Z p Um,n ab
× Kakde’s additive conditions (A1)-(A3). Analogously, ⊂ m Z p Um,n consists of those elements ym satisfying the multiplicative conditions (M1)-(M4) below, which we have specialised from [7, p107] to our particular situation: (M1) Nm−1,m ym−1 = πm,m−1 ym for all m ≥ 1; (M2) ym = gym g −1 at every g ∈ G∞,n; for each m ≥ 1; (M3) ym ≡ Verm−1,m (ym−1 ) mod Im σ. (ν) p ! (ν) p "m ym ym−1 (ν) − ϕ (ν) ∈ p · Im σm(ν) for every m ≥ 0. (M4) Nm,m+1 ym
123
Nm−1,m ym−1
K1 -congruences for three-dimensional
ab
ab
Here in condition (M3), the homomorphism σ. → Z p Um,n denotes the m : Z p Um,n ( p−1 − pm−1 i p m−1 i . additive map sending f → i=0 γ f γ Warning: If a sequence ym satisfies conditions (M1)–(M4), then its image under L automatically satisfies by [7, p107, Lemma 4.5]. Unfortunately, because the family ab(A1)–(A3) of abelianizations Um,n we use is coarser than that considered in [7,17], we cannot 0≤m≤n−s directly apply the results in op. cit. to obtain a converse statement such as ab
ab
× ? L (ym ) ∈ Z p Um,n ⇒ ym ∈ Z p Um,n . (A1)-(A3)
m
(M1)-(M4)
m
The salvage is to show that K 1 Z p [[G∞,n ]] splits into a direct product of K 1 Z p [[]] with with a complementary factor W† ; we shall then construct a section S : p · → ∞,n W† for which L ◦ S and S ◦ L (W ) are both identity maps. One concludes that ym arises ∞,n †
Z [[G ]] if and only if L (ym ) ∈ p·, which is itself equivalent to the sequence from K p ∞,n 1 χ ◦ L (ym ) satisfying constraints (C1)–(C4) from Theorem 15.
4.1 Convergence of the logarithm on Im(σm ) We will shortly introduce the Taylor-Oliver logarithm, which is usually defined in terms of group algebras arising from finite groups. Since the profinite groups G∞,n and Um,n are both infinite, one should instead consider their finite counterparts ν
pn
ν
pn
(ν) (ν) G∞,n := / p H∞ /H∞ and more generally Um,n := p / p H∞ /H∞ , m
(ν)
(ν)
at each integer triple m, n, ν ∈ Z with 0 ≤ m ≤ n − s ≤ ν. For example, U0,n equals G∞,n . ab ∼ Remark Using Proposition 7, one has Un−s,n = Un−s,n ; in other words Un−s,n is abelian. It ν pn p follows that acts trivially on H∞ /H∞ for all ν ≥ n − s, so the semi-direct products above make good sense. Whenever we write the superscript (ν) above an object or a map, we mean the analogue of that object/map for the corresponding finite group (providing the object/map descends to its finite version, of course). m
Now recall from Proposition 10(ii) that Im(σm ) is freely generated over Z p [[ p ]] by the (m,n) (ν) (m,n) elements A with ∈ orb H∞ . It is therefore trivially true that Im σm must be m (m,n) ν
(m,n) generated over Z p p / p by the same A ’s. If 1 , 2 ∈ orb H∞ contain h 1 and h 2 respectively, then A(m,n) 1
· A(m,n) 2
=
m −1 p
γ
−i
h1γ · i
i=0
=
m −1 p
t=0
m −1 p
γ
−j
h2γ = j
j=0
A
m −1 p m −1 p
i=0
γ j−i i γ −i h 1 h 2 γ
j=0
(m,n) γt
h1 h2
(ν), ab
(ν) (ν) which belongs to the image of σm . It follows that Im σm is an ideal of Z p Um,n . Iterating the above calculation N -times, one deduces that A(m,n) 1
(m,n) · A(m,n) 2 · · · A N +1
=
m −1 p m −1 p
t1 =0 t2 =0
···
m −1 p
t N =0
A
(m,n)
γ t1
γ tN
h 1 h 2 ···h N +1
123
D. Delbourgo, Q. Chao
(m,n) and element h ∈ , which means for each ∈ orb H∞
A(m,n)
N +1
=
m −1 p
t1 =0
···
m −1 p
A
t N =0
(m,n) hh
γ t1
···h
γ tN
=
N +1 j=2
pm (m,n) · ··· A . hw2 ···w N +1 # w ∈ w ∈ N +1
2
N +1
(ν) (ν) ∈ p N · Im σm ⊂ p · Im σm . m • Alternatively, if # = p m so that Stab/ pm (h) = γ p , then N +1 (m,n) (m,n) A(m,n) = ··· A = A γ t1 (m,n)
• Clearly if # < p m , then A
w2 ∈
w N +1 ∈
hw2 ···w N +1
(t1 ,...,t N )∈(Z/ p m Z)⊕N
γ t1
hh
···h
γ tN
.
γ tN
There are at most p m N distinct elements of the form h h · · · h , whilst the total number (m,n) of elements in H∞ is p 2s+2m+, p if () =(II), where by Proposition 7 the term ⎧ ⎪ in Cases (III), (IV) ⎨0 (m) (m) , p := N,1 + N,2 − 2s − 2m = in Case (V) ord p (d) ⎪ ⎩ r + ord p (t) in Case (VI) is independent of m and n.
γ t1
γ tN
will start repeating, Consequently for m N ≥ 2s +2m +, p these elements h h · · · h (ν) (m,n) N +1 in which case A ∈ p · Im σm . Note that the latter inequality is equivalent to N +1≥3+
2s+, p , m
so we arrive at the following estimate: , + (m,n) j log( j) j A
j
∈ p
3+
2s+, p m
− log( p)
· Im σm(ν) .
(15)
If one sets , p = −s and n = m, a similar argument implies (15) also holds for () =(II). (∞ j j+1 y and Proposition 19 (a) The two formal power series log(1 + y) = j=1 (−1) j (ν) ( j j (1 + y)−1 = ∞ j=0 (−1) y converge for all y ∈ Im σm . / 2s+, p 0 3+ m then for every N ≥ 1, the logarithm induces a natural isomorphism (b) If δm := p (ν) δ ·N (ν) δ ·N 1 + Im σm m Im σm m ∼ log : (ν) δ ·N +1 −→ (ν) δ ·N +1 ; 1 + Im σm m Im σm m in particular, if p ≥ 5 and one chooses m ≥ 2s + , p , then δm = 1 above. (ν) (ν) log (ν) exp (c) There are isomorphisms 1 + p · Im σm → p · Im σm and p · Im σm → 1 + p · (ν) Im σm which are mutually inverse maps to one another. Proof To show (a) one uses the estimate (15) together with the fact that the exponent , + j log( j) j − log( → ∞ as j → ∞, which implies both lim j→∞ (−1) j+1 yj = 0 and 2s+, p p) 3+ m (ν) j (ν) ⊂ p · Im σm for j 0, the topology lim j→∞ (−1) j y j = 0. In fact, since Im σm (ν) j induced by the neighborhoods Im σm coincides with the p-adic topology. j∈N The assertion in (c) can be proved by following an identical argument to [7, p106], which leaves us to tackle (b).
123
K1 -congruences for three-dimensional (m,n) p (ν) For simplicity we suppose that p ≥ 5 and m ≥ 2s + , p , so that (Ap ) ∈ Im σm by (ν) (ν) p the estimate (15), whence yp ∈ Im σm for all y ∈ Im σm . Consider the homomorphism
N log : 1 + Im σm(ν) → †
(ν) N Im σm (ν) N +1 ⊗Z p Q p Im σm
(ν) N +1 given by log† (1+ y) := log(1+ y) mod Im σm . Assuming that j > 1, let us examine (ν) N yj j+1 : the p-integrality of (−1) j for each y = a1 · · · a N ∈ Im σm j
(ν) N j (ν) N +1 a1 ···a N ∈ Im σm ⊂ Im σm ; j p (ν) 1+ p(N −1) (ν) N +1 a1 p p ⊂ Im σm ; p · a2 · · · a N ∈ Im σm
• If p j then (−1) j+1 yj = ± p
j
j
• If j = p then (−1) p+1 yp = • If j = p k with k > 1, then p k pk k+ pk N − pk N +1 a1 p k − pk pk pk p k +1 y = · a1 · a2 · · · a N ∈ Im σm(ν) ⊂ Im σm(ν) . (−1) k p p Lastly, the general case where j = p k c with p c and j > 1 reduces to the previous cases, upon replacing y with y c throughout. (ν) N +1 (ν) N j for every y ∈ Im σm and j > 1. We therefore conclude (−1) j+1 yj ∈ Im σm (ν) N N (ν) (ν) → Im(σ(ν)m N) +1 Because log† (1 + y) ≡ y mod Im(σm ) N +1 , clearly log† : 1 + Im σm Im(σm ) (ν) N +1 must be a surjective map; further, one easily checks that 1 + Im σm ⊂ Ker log† . Assertion (b) now follows immediately for p ≥ 5 and m ≥ 2s + , p . Finally, to treat assertion (b) when p = 3 or m < 2s + , p , one simply observes that if 2s+ (ν) (ν) δm p 3+ m, p then (y p ) ∈ Im σm for all y ∈ Im σm , using the estimate (15) again. δm ≥ p One then repeats the previous arguments, with y replaced by y δm everywhere.
4.2 Interaction of the theta-maps with both ϕ and log We now derive some technical results describing how the Frobenius mapping ϕ and the logarithm commute with the theta-homomorphisms. Let us recall that in our situation, the (ν) (ν) trace and norm maps from G∞,n down to Um,n have the simple description TrG (ν)
(ν) ∞,n /Um,n
(α) =
m −1 p
γ −k αγ k
and
NormG (ν)
(ν) ∞,n /Um,n
k=0 (ν),+
Definition 20 (a) The additive theta-map θm,n given by the composition (ν),+ θm,n (−) := TrG (ν)
(ν) ∞,n /Um,n
(x) =
m −1 p
γ −k xγ k .
k=0
(ν)
(ν), ab
: Z p Conj G∞,n → Z p Um,n is
(ν) (ν)
(−) mod Um,n , Um,n .
(ν) (ν), ab × (ν) (b) The multiplicative theta-map θm,n : K 1 Z p G∞,n → Z p Um,n is defined by (ν) (−) := NormG (ν) θm,n
(ν) ∞,n /Um,n
(ν) (ν)
(−) mod Um,n , Um,n .
123
D. Delbourgo, Q. Chao
(ν)
ν
Let ι : Z p / p → Z p G∞,n be the map on group algebras induced from the sequence ν
∼
(ν)
ν
ν
/ p → / p {1} → G∞,n that identifies / p with a non-normal subgroup of (ν) G∞,n . Lemma 21 There exists a splitting of abelian groups (ν) ∼ ν × (ν) −→ Z p / p × W† sending x → x cy , x † , K 1 Z p G∞,n (ν)
where x cy = ι∗ ◦ θ0,n (x), x † =
(ν) x x cy , and the complement W† :=
(ν) x † x ∈ K 1 Z p G∞,n .
(ν) (ν)
pn (ν) Proof Firstly θ0,n coincides with the quotient mapping modulo U0,n , U0,n = H∞ /H∞ . The composition / p
ν
ι
(ν)
→ G∞,n
mod H∞ / p n
ν
/ p equals the identity, and this induces
(ν) (ν) θ0,n ν ν ι∗ K 1 Z p / p −→ K 1 Z p G∞,n −→ K 1 Z p / p
ν ν × ∼ which must then be the identity map on K 1 Z p / p = Z p / p . The latter group (ν) is therefore isomorphic to a direct factor of K 1 Z p G∞,n , and the rest follows easily. For a group G, the ring homomorphism ϕG : Z p [ Conj(G)] → Z p [ Conj(G)] denotes the linear extension of the map [g] → [g p ] on Conj(G) (note if G is abelian, then Conj(G) = G). (ν)
Lemma 22 For all α ∈ Q p Conj G∞,n ,
(ν),+ θm,n
◦ ϕG (ν) (α) = ∞,n
⎧ ⎨p · ϕ ⎩ϕ
(ν) (ν)
◦ TrG (ν) /U (ν) (α) mod Um,n , Um,n if m ≥ 1 ∞,n m−1,n (ν) (ν)
if m = 0. (α) mod U0,n , U0,n (ν)
Um−1,n
(ν)
G∞,n
Proof If m = 0, the formula is straightforward to establish. We therefore suppose that m ≥ 1. It is enough to consider conjugacy classes of the form (ν)
α = [γ j · h] with j ∈ Z/ p ν Z and h ∈ Hp∞n , since these will generate Q p Conj G∞,n . H∞
Key Claims:
p p−1 γ ji ν (I) For all j ∈ Z/ p ν Z, one has γ j · h = γ pj · i=0 h inside / p (II) If
k, k
∈ Z satisfy k ≡ ϕU (ν)
m−1,n
123
γj ·h
k
( mod
γk
p m−1 ),
≡ ϕU (ν)
m−1,n
H∞ pn H∞
.
then
γj ·h
γk
(ν)
(ν) , Um,n mod Um,n .
(16)
K1 -congruences for three-dimensional
Postponing their proof for the moment, one calculates that ⎛ ⎞ p−1 1 γ ji 2 by (I) (ν),+ (ν),+ ⎝ ⎠ γ pj · θm,n ◦ ϕG (ν) [γ j · h] = θm,n h ∞,n
i=0
⎧ ⎨γ pj · ( pm −1 γ −k p−1 h γ ji γ k mod U (ν) , U (ν) if γ pj ∈ pm m,n m,n k=0 i=0 = ⎩0 otherwise ji
m ( m−1 p −1 p−1 −k γ (ν) (ν) γ pj · k=0 h γ k mod Um,n , Um,n if γ j ∈ p i=0 γ = 0 otherwise ⎧
m −1 γ k ( m−1 p (ν) (ν) ⎨ γ j · k=0 h mod Um,n , Um,n if γ j ∈ p by (I) ϕU (ν) m−1,n = ⎩0 otherwise ⎧
k (ν) (ν)
( pm−1 −1 γ m−1 ⎨ γ j · p · k =0 h mod Um,n , Um,n if γ j ∈ p by (II) ϕ (ν) Um−1,n = ⎩0 otherwise (ν) (ν)
j [γ · h] mod Um,n , Um,n . = p · ϕU (ν) ◦ TrG (ν) /U (ν) ∞,n
m−1,n
m−1,n
(ν)
The full lemma now follows for each m ≥ 1, as Q p Conj G∞,n is generated by [γ j · h]’s. γj
It remains to establish Claims (I) and (II). To prove (I) we know that h · γ j = γ j · h , in which case j p γj = γ j · h · γ j · h · γ j · h ··· γ j · h = γ2j · h · h · γ j · h ··· γ j · h γ ·h γj γj γ2j γj = γ2j · h · γ j · h · h · · · γ j · h = γ3j · h · h · h ··· γ j · h = . . . = γ ( p−1) j · h
γ ( p−2) j
·h
γ ( p−3) j
p−1 γ ji · · · h · γ j · h = . . . = γ pj · h . i=0
by (I)
p−1
γk
p−1
γ ji+k
To show (II) note that the L.H.S. of (16) = γ pj · i=0 (h )γ = γ pj · i=0 h , ji+k
γ p−1 by an identical argument; one deduces that while the R.H.S. of (16) = γ pj · i=0 h ⎞ ⎛ p−1 γ ji+k −1 ji+k
L.H.S. of (16) ⎠ · γ − pj = γ pj · ⎝ h (h )γ R.H.S. of (16) i=0 ⎞ ⎛ p−1
−1 · γ ji+k ⎠ · γ − pj . = γ pj · ⎝ γ −( ji+k ) · γ k −k · h · γ −(k −k) · h ji
i=0
(ν)
m−1 (ν) ∈ Um−1,n , Um−1,n because γ k−k ∈ p
− pj p−1 γ ji+k γ L.H.S. of (16) whenever k ≡ k ( mod p m−1 ), which in turn implies R.H.S. . i=0 h k,k
of (16) = This latter product is divisible by p, in fact
p
(ν) (ν) L.H.S. of (16) (ν) (ν) . ⊂ Um,n , Um,n ∈ Um−1,n , Um−1,n R.H.S. of (16) (ν) (ν)
Therefore L.H.S. ≡ R.H.S. mod Um,n , Um,n , which establishes Claim (II) as well.
However h k,k := γ k −k · h · γ −(k −k) · h
−1
123
D. Delbourgo, Q. Chao (ν)
We now examine how the Frobenius map ϕ commutes with θm−1,n . Consider the sequence m−1
m
m
p p (H∞ ) p p (H∞ ) p H∞ (−) p × −→ × ×
ν ν ν p (ν) (ν) (ν) (ν) (ν) (ν)
p p p Um−1,n , Um−1,n Um−1,n , Um−1,n Um,n , Um,n
p (ν) (ν)
(ν) (ν) induced from the p-power map, and the containment Um−1,n , Um−1,n → Um,n , Um,n . (ν), ab
(ν), ab
ϕ one obtains If we label the composition as 3 ϕ : Um−1,n → Um,n , by linearly extending 3 (ν), ab
(ν), ab
3 ϕU (ν), ab : Q p Um−1,n → Q p Um,n , cg · [g] → cg · 3 ϕ [g] m−1,n
(ν), ab
(ν), ab
g∈Um−1,n
g∈Um−1,n
as a homomorphism of commutative algebras. (ν) Lemma 23 (i) For each integer m ≥ 1 and every x ∈ K 1 Z p G∞,n , 3 ϕU (ν), ab ◦ logZ m−1,n
(ν), ab p [Um−1,n ]
(ν)
◦ θm−1,n (x)
! = ϕU (ν), ab logZ m−1,n
(ν), ab p [Um−1,n ]
◦ NormG (ν)
(ν) ∞,n /Um−1,n
" (x)
(ν)
(ν) mod Um,n , Um,n .
(ν) (ii) For each integer m ≥ 0 and every x ∈ K 1 Z p G∞,n , † (ν) θm,n x =
(ν) θm,n (x) (ν) (m,ν) τ∗ ◦ N0,m θ0,n (x)
and
cy (ν) (ν) x = τ∗(m,ν) ◦ N0,m (θ0,n (x)) θm,n
m (ν), ab
ν
where τ (m,ν) denotes the natural inclusion Q p p / p → Q p Um,n . At first glance these statements are rather technical in nature, and their demonstrations could easily be skipped on an initial reading. However they will become important tools for us in the next section, when we calculate the Taylor-Oliver logarithm composed with the (ν),+ family of theta-maps θm,n 0≤m≤n−s . (ν)
p (ν) (ν) (ν)
Proof Starting with assertion (i), since Um−1,n , Um−1,n ⊂ Um,n , Um,n one deduces
(ν) (ν) ϕU (ν), ab ◦ TrG (ν) /U (ν) (α) mod Um,n , Um,n ∞,n m−1,n m−1,n ! "
(ν) (ν) (ν),+ = 3 ϕU (ν), ab TrG (ν) /U (ν) (α) mod Um−1,n , Um−1,n = 3 ϕU (ν), ab ◦ θm−1,n (α) ∞,n
m−1,n
m−1,n
m−1,n
(17) (ν)
for every α ∈ Q p Conj G∞,n . Evaluating both sides at α = log(x), it is easily verified ϕU (ν), ab ◦ log ◦ NormG (ν) /U (ν) (x) ≡ ϕU (ν), ab ◦ TrG (ν) /U (ν) log(x)
∞,n
m−1,n
by (17)
=
m−1,n
3 ϕU (ν), ab m−1,n
m−1,n
(ν),+ ◦ θm−1,n log(x)
∞,n
m−1,n
(ν)
= 3 ϕU (ν), ab ◦ log ◦ θm−1,n (x). m−1,n
To prove (ii), one simply observes that (ν) pn τ∗(m,ν) ◦ N0,m θ0,n (x) = τ∗(m,ν) ◦ Norm/ pm x mod H∞ /H∞ (ν) (ν)
pn = NormG (ν) /U (ν) τ∗(0,ν) x mod H∞ /H∞ mod Um,n , Um,n m,n ∞,n n p (ν) (ν) x cy . ◦ ι∗ x mod H∞ /H∞ = θm,n = θm,n
123
K1 -congruences for three-dimensional (ν) Consequently θm,n x † =
(ν)
θm,n (x)
(ν) θm,n (x cy )
=
(ν) θm,n (x) (m,ν) (ν) τ∗ ◦N0,m (θ0,n (x))
, and the two identities follow.
4.3 The image of the Taylor-Oliver logarithm
For a finite group G, the Taylor-Oliver logarithm LOGG : K 1 Z p [G] → Z p Conj(G) is defined by LOGG (x) := logZ p [G] (x) −
1 ϕG logZ p [G] (x) p
where logZ p [G] is the unique extension of log Jac(Z p [G]) (see [22] for more details). Note that G need not necessarily be a p-group, even though it happens to be so in this paper. (ν) If G = G∞,n then LOGG (ν) denotes the ν-th layer of the map ‘ LOG’ occurring in (14). ∞,n Our task is to calculate the mappings L and Lχ which make that diagram commutative. The former of these maps may be determined from the following formulae. (ν) Proposition 24 (a) If m ∈ {1, . . . , n − s} and x ∈ K 1 Z p [[G∞,n ]] , then ⎛ ⎞ (ν) (x) θ m,n (ν),+ ⎠. ◦ LOGG (ν) (x) = logZ [U (ν), ab ] ⎝ θm,n (ν) p m,n ∞,n 3 ϕU (ν), ab ◦ θm−1,n (x) m−1,n
(ν) W†
= x cy ∈ then (ν),+ ◦ LOGG (ν) x † θm,n ∞,n (m−1,ν) (ν)
(ν) τ∗ ◦ N0,m−1 θ0,n (x) θm,n (x) = logZ [U (ν), ab ] ϕU (ν), ab . (ν) · 3 (m,ν) (ν) p m,n m−1,n τ∗ ◦ N0,m θ0,n (x) θm−1,n (x)
(b) Furthermore, if
x†
x
Proof Using the definition of the Taylor-Oliver logarithm and our previous results, 1 (ν),+ ◦ ϕG (ν) logZ [G (ν) ] (x) ·θ p ∞,n ∞,n p m,n (ν) (ν)
1 by22 (ν),+ , Um,n = θm,n log(x) − · p · ϕU (ν) ◦ TrG (ν) /U (ν) log(x) mod Um,n ∞,n m−1,n m−1,n p (ν) (ν)
(ν),+ log(x) − ϕU (ν) ◦ log NormG (ν) /U (ν) (x) mod Um,n , Um,n = θm,n
(ν),+ (ν),+ ◦ LOGG (ν) (x) = θm,n ◦ logZ θm,n ∞,n
(ν) p [G∞,n ]
(x) −
∞,n
m−1,n
m−1,n
by23(i) (ν) = θm,n logZ [G (ν) ] (x) − 3 ϕU (ν), ab ◦ logZ [U (ν), ab ] ◦ θm−1,n (x) p ∞,n p m−1,n m−1,n (ν) (ν) ϕU (ν), ab ◦ θm−1,n (x) (x) − logZ [U (ν), ab ] 3 = logZ [U (ν), ab ] θm,n (ν),+
p
p
m,n
m,n
m−1,n
which establishes assertion (a). To prove (b), one simply combines part (a) with the formula from Lemma 23(ii).
Remark As a direct consequence, in order to make the left-hand square in the diagram (ν) K 1 Z p [G∞,n ] ⏐ LOG
(ν) G∞,n
(ν) Z p Conj(G∞,n )
(ν)
θm,n
−→
χ (ν) −→ χ (ν) ⏐ ⏐ (ν) (ν) L
(ν),+
θm,n
−→
Lχ
χ (ν) −→ χ (ν)
123
D. Delbourgo, Q. Chao
commutative, it follows from Proposition 24(a) that one should define ⎞ ⎛ (ν) (ν) (ν), ab × y m (ν) L(ν) ym := logZ [U (ν), ab ] ⎝ Z p Um,n . (ν) ⎠ for all ym ∈ p m,n m 3 ϕU (ν), ab ym−1 0≤m≤n−s m−1,n
(18) (ν) To make the right-hand square commutative, we need to work out the map Lχ explicitly. Fix (m,n)
a finite order character χ : H∞ → μ p∞ factoring through the quotient group H∞ , which one may interpret as a homomorphism ν m ν (m,n) (ν), ab ∼ p m χ : Um,n = / p × H∞ −→ p / p × Im(χ) (ν), ab
satisfies sending an element γ j · h to γ j · χ(h). It follows that its extension to Z p Um,n ⎛ ⎞
(ν),+ χ θm,n ◦ LOGG (ν) (x) = log ∞,n
⎜ pm ⎜ ⎝ Oχ pν
(ν) ⎟ χ ◦ θm,n (x) ⎟ ⎠. (ν) ϕ pm−1 χ p ◦ θm−1,n (x) p
ν
(ν)
Moreover by Proposition 24(b), for any x † = x/x cy ∈ W† one has (ν)
(ν) N0,m−1 θ0,n (x) † χ ◦ θm,n (x) (ν),+
= log χ θm,n ◦ LOGG (ν) x pm (ν) · ϕ pm−1 (ν) ∞,n Oχ pν ν N0,m θ0,n (x) χ p ◦ θm−1,n (x) p (ν) (ν) m ν as χ acts trivially on Z p [ p / p ], and thus also on N0,m−1 θ0,n (x) and N0,m θ0,n (x) . (ν) (ν) Since ym,χ corresponds to χ ◦ θm,n (x), the preceding formulae imply one should define ⎛ ⎞ 4 m 5× (ν) ⎜ ⎟ (ν) (ν) ym,χ p (ν) ⎜ ⎟
m Lχ (ym,χ ) m,χ := log O . ∈ where y χ ν m,χ p ⎝ ⎠ (ν) Oχ ν p ϕ pm−1 ym−1,χ p p m,χ
(ν)
Indeed if ym,χ ∈
(ν) L(ν) χ (ym,χ ) m,χ (ν)
ym,χ
p
(ν)
(ν)
χ ◦ θm,n W† = log pm
m,χ
Oχ
∈ 1 + p · Oχ
p
ν
ν
, then one can further say (ν)
(ν) N0,m−1 y0,1 ym,χ . (ν) · ϕ pm−1 (ν) ν N0,m y0,1 ym−1,χ p p
pm
(19)
for all m, so the full expression occurring inside the pm
logarithm in Eq. (19) must automatically be congruent to 1 modulo p · OC p pν .
In fact
(ν) N0,m y0,1
p
ν
(ν) (ν) (ν) (ν) (ν) Corollary 25 If ym ∈ ∞,n W† and one sets ym,χ = χ (ym ) , then both (ν) L(ν) (ym ) ∈ (ν) ∩ p · Im σm(ν) and m (ν) (ν) m ν
∩ p· L(ν) OC p p / p . χ (ym,χ ) ∈ χ m,χ
Proof To address the first assertion, Proposition 24(b) implies that (ν)
(ν) (ν) N0,m−1 y0 ym (ν) (ym ) m = logZ [U (ν), ab ] L ϕU (ν), ab (ν) · 3 (ν) p m,n m−1,n N0,m y0 ym−1
123
K1 -congruences for three-dimensional (ν)
and as each of the two fractions inside the logarithm belongs to the group 1 + p · Im(σm ), the containment follows directly from Proposition 19(c). To establish the second assertion, one combines the discussion after Eq. (19) together with m m ν ∼ ν
the isomorphism log : 1 + p · OC p p / p −→ p · OC p p / p .
4.4 A proof of Theorems 1 and 2 (ν) Recall from earlier that if a sequence ym satisfies conditions (M1)-(M4), then its image under L(ν) always satisfies (A1)-(A3). We shall now establish a converse statement (ν) (ν) L(ν) (ym ) ∈ p · (ν) ⇒ ym ∈ (ν) . (ν) (ν) If we are successful, the question as to whether or not ym arises from K 1 Z p [G∞,n ] (ν) (ν) (ν) under the mapping ∞,n reduces to determining whether or not Lχ (ym,χ ) ∈ χ (ν) . To achieve this goal, we will explicitly construct a section ⎞ ⎞ ⎛ ⎛
(ν), ab ⎠ (ν), ab ⎠ S (ν) : ⎝ p · Z p Um,n −→ ⎝ 1 + p · Z p Um,n 0≤m≤n−s
0≤m≤n−s
(A1)-(A3)
for which L(ν) ◦ S (ν)
and S (ν) ◦ L(ν)
(M1)-(M4)
are the respective identity mappings. (ν) (ν), ab
let us first fix a sequence am ∈ 0≤m≤n−s p · Z p Um,n . To produce this (ν), ab ∼ (ν), ab
−→ 1 + p · Z p Um,n is an isomorphism of abelian Recall that exp : p · Z p Um,n groups. p· (ν) map S (ν) ,
(ν)
(ν)
∞,n (W† )
(ν) (ν) Definition 26 Given the sequence am above, one recursively defines y0 := 1 and (ν) (ν) ym := 3 ϕU (ν), ab ym−1 × expZ m−1,n
(ν), ab p [Um,n ]
(ν) am
for each m ≥ 1,
(ν), ab
(ν) (ν) so that ym ∈ m 1 + p · Z p Um,n . We label this association am → ym by S (ν). (ν), ab
Lemma 27 (i) The composition L(ν) ◦ S (ν) is the identity map on m p · Z p Um,n . (ν), ab
(ii) The composition S (ν) ◦ L(ν) yields the identity map on m 1 + p · Z p Um,n . Proof To establish the first assertion, one simply calculates that (ν) (ν) by(18) L(ν) ◦ S (ν) (am ) m = L(ν) (ym ) = logZ by26
=
logZ [U (ν), ab ] expZ p
m,n
(ν) am (ν), ab p [Um,n ]
⎞ (ν) y m (ν), ab ⎝ (ν) ⎠ p [Um,n ] 3 ϕU (ν), ab ym−1 ⎛
m−1,n
=
(ν) . am
The proof of the second assertion follows along identical lines.
(ν), ab
(ν) satisfies (A1)– For the rest of this section, we assume that am ∈ m p · Z p Um,n (ν) (ν) (A3). The goal now is to prove that properties (M1)–(M4) all hold for ym = S (ν) (am ) . Three of them are straightforward to deduce, but property (M3) requires more effort.
123
D. Delbourgo, Q. Chao
(ν) Establishing that S (ν) (am ) satisfies (M1),(M2),(M4). Let us begin by obtaining (M1). (ν) Since (A1) holds for the sequence am , clearly (ν) (ν) Nm−1,m ◦ expZ [U (ν), ab ] am−1 = expZ [U (ν), ab ] ◦ Trm−1,m am−1 p
p
m−1,n
m,n
(ν) (ν) = expZ [U (ν), ab ] ◦ πm,m−1 am = πm,m−1 ◦ expZ [U (ν), ab ] am p m,n p m,n (ν) (ν) Nm−1,m ym−1 πm,m−1 ym (ν) = (ν) for each m ≥ 1. The latter is equivalent to i.e. by (A1)
Nm−1,m 3 ϕ (ym−2 )
πm,m−1 3 ϕ (ym−1 )
(ν) Nm−1,m ym−1
=
(ν) πm,m−1 ym
× 3 ϕU (ν), ab m−1,n
(ν)
Nm−2,m−1 ym−2
(ν) πm−1,m−2 ym−1
.
(ν) (ν) The equality between Nm−1,m ym−1 and πm,m−1 ym now follows by induction on m, thereby yielding (M1) as a consequence. ν (ν) Focussing instead on (M2), the semi-direct product structure on G∞,n = / p Hp∞n (ν)
(ν), ab
H∞
implies the subset of G∞,n -invariant elements in Z p [Um,n ] consists of (ν)
(ν), ab
(ν), ab (ν), ab . , Z p [Um,n ] = H 0 , Z p [Um,n ] = Im σm(ν) 1/ p ∩ Z p Um,n H 0 G∞,n
(ν)
(ν) (ν) (ν), ab Now (A2) states that am belongs to this subset, hence ym ∈ Im σm 1/ p ∩Z p [Um,n ]× upon combining the recurrence in Definition 26 with induction on m, and (M2) follows. (ν), ab
To show that (M4) holds true, consider the trace mapping Trm,m+1 acting on Z p Um,n . For each integer m ≥ 0, one may decompose 1 m+1 ν 2 (ν), ab
(m,n) ∼ p × H∞ ⊕ Ker Trm,m+1 Z p Um,n = Zp p where by Lemma 14, the trace acts through multiplication by p on the first factor and kills off the second factor. (ν), ab
(ν) (ν) (ν) Note that am ∈ p · Z p Um,n so 1p Trm,m+1 am ≡ am mod p · Ker Trm,m+1 . (ν) (ν) (ν) (ν) Moreover the sequence am satisfies (A3), thus p · am − Trm,m+1 am ∈ p · Im σm and applying Proposition 19: (ν) (ν) − Trm,m+1 am ∈ 1 + p · Im σm(ν) . expZ [U (ν), ab ] p · am p
m,n
(ν) (ν) (ν) exp(am ) p It is easy to see exp p · am − Trm,m+1 (am ) = (ν) . Also, recalling from Nm,m+1 ◦ exp(am ) (ν) (ν) earlier that exp am = ym(ν) , we therefore conclude
3 ϕ (ym−1 )
⎞−1 ⎛ (ν) (ν) p (ν) ym Nm,m+1 ym (ν) p × ⎝ (ν) ⎠ ∈ 1 + p · Im σm . 3 ϕU (ν), ab ym−1 Nm,m+1 ◦ 3 ϕU (ν), ab ym−1 m,n
m−1,n
Equivalently
! (ν) p ym (ν) × 3 ϕU (ν), ab
Nm,m+1 ym
m−1,n
(ν)
ym−1
p (ν)
Nm−1,m ym−1
"−1 (ν) ∈ 1 + p · Im σm , so (M4) holds.
(ν) Establishing that S (ν) (am ) satisfies (M3). We begin with a technical result describing the (ν), ab
(ν), ab
image of the map σ3m(ν) : Z p Um,n → Z p Um,n sending ( p−1 m−1 m−1 f → i=0 γ − p i f γ p i .
123
K1 -congruences for three-dimensional
Lemma 28 For each m ∈ {0, . . . , n − s}, the -invariant submodule H 0 , Im σ3m(ν) is
ν finitely generated over Z p / p by the combined set (m,n) , # = p m A(m,n) ∈ orb H∞ $ % (m,n) # (m,n) m , # < p ∪ · A ∈ orb H∞ p m−1 (ν) and in particular, Im σm ⊂ H 0 , Im σ3m(ν) ⊂ Im σ3m(ν) . m
(m,n)
ν
Proof Because a generator γ ∈ acts trivially on p / p and through I2 + M on H∞ , (ν), ab m (m,n) ν
= Z p p / p ⊗Z p H 0 I2 + M , Z p H∞ H 0 , Z p Um,n 6 7 pm
(m,n) pν · h ∈ orb H∞ = Zp / 8
h ∈
9 pm
# (m,n) pν (m,n) · · A = Zp / ∈ orb H∞ pm ( pm −1 (
(m,n) pm · h ∈ h . where we have employed the basic identity Ah = i=0 γ −i hγ i = # h h (m,n) (
# (m,n) Now pick an element pmh · Ah = h ∈ h belonging to H 0 I2 + M , Z p H∞ . h Then one easily sees that p −1 p−1 #h #h − j j (m,n) · A = · γ hγ = h pm pm m
j=0
i=0
p m−1 −1 j=0
m−1 #h m−1 · γ − p i γ − j hγ j γ p i m p
(m,n)
# ( p m−1 −1 − j which coincides exactly with σ3m(ν) f h , where f h := pmh · j=0 γ hγ j ∈ Q p H∞ . (m,n)
# (m,n) ∈ Im σ3m(ν) if and only if p z · f h ∈ Z p H∞ , and as It follows that p z · pmh · Ah ⎧ ⎨ p z · ( pm−1 −1 γ − j hγ j if # = p m h z j=0 p · fh = (
m ⎩ p z−1 ·
h if # h < p , h ∈ h
z ≥ 1if # < p m . the latter condition occurs when z ≥ 0 if # =p m , or alternatively m Therefore the union of the sets f h #h = p and p · f h #h < p m will generate ν
the -invariant part of Im σ3m(ν) over Z p / p , as asserted. (ν) (m,n) Finally, the inclusion Im σm → H 0 , Im σ3m(ν) occurs as the generators A of the left-hand module are p-integral multiples of generators for the right-hand module. Verm−1,m
Proposition 29 For each m ≥ 1, the transfer sends p · Im(σm−1 ) −→ x y
Im σ3m(ν) .
(m−1,n)
and f (X ) ∈ Z p [[X ]], then from Lemma 18: Proof If we choose any h = h 1 h 2 ∈ H∞ m−1 m (m,n) (m−1,n) = p −1 × f γ p − 1 · A x y
−1 ·A x y Verm−1,m f γ p h1 h2
h1 h2
! " x where ∈ Z2p is given in Lemma 12. Setting f (X ) = p, it follows immediately that y by 28 (m−1,n) (m,n) Verm−1,m p · A x y = A x y ∈ Im σm(ν) → Im σ3m(ν) . h1 h2
h1 h2
123
D. Delbourgo, Q. Chao
(ν) Lastly applying Proposition 10(ii), we know p· Im σm−1 is freely generated over the algebra m−1 ν
(m−1,n) Z p p / p by the set of p · A x y ’s, hence the result is proven. h1 h2
(ν) (ν) Let us now establish that (M3) holds for ym = S (ν) (am ) . For each integer m ≥ 2, (ν) (ν) (ν) 3 ϕU (ν), ab ym−1 × expZ [U (ν), ab ] am ym by 26 p m,n m−1,n (ν) = (ν) (ν) Verm−1,m ym−1 ϕU (ν), ab ym−2 × expZ [U (ν), ab ] am−1 Verm−1,m 3 p m−1,n m−2,n
(ν) (ν) ym−1 (ν) am = 3 ϕU (ν), ab − Verm−1,m am−1 (ν), ab (ν) × expZ p [Um,n ] m−1,n Verm−2,m−1 ym−2 (ν) (ν) and the term am − Verm−1,m am−1 ∈ Im σ3m(ν) , using Lemma 28 and Proposition 29. An identical argument to Proposition 19(b) shows that expZ
(ν), ab p [Um,n ]
:
Im(σ3m(ν) ) N Im(σ3m(ν) ) N +1
∼
−→
1 + Im(σ3m(ν) ) N 1 + Im(σ3m(ν) ) N +1
is an isomorphism for every N ≥ 1, in which case
(ν) (ν) ym−1 ym ϕU (ν), ab (ν) = 3 (ν) × 1 + dm m−1,n Verm−1,m ym−1 Verm−2,m−1 ym−2 (ν) for some dm ∈ Im σ3m . (ν) ⊂ Im σ3m(ν) . Furthermore, one easily checks the containment 3 ϕU (ν), ab Im σ m−1 m−1,n (ν) ym−1 (ν) (ν) ∈ 1 + Im σ , one may conclude Therefore, if we inductively assume m−1 Verm−2,m−1 ym−2
(ν)
ym
(ν)
Verm−1,m ym−1
∈ 1 + Im σ3m(ν) . Property (M3) then follows for all m ≥ 2 by induction. (If
m = 1 the same argument works fine, except one omits the denominator terms above.) Proof of Theorem 2. As mentioned earlier, now that we have constructed the section S (ν) (ν) (ν) mapping p · (ν) into (ν) , to check whether ym arises from an element of K 1 Z p [G∞,n ] (ν) (ν) it is the same as verifying if Lχ (ym,χ ) ∈ χ (ν) . However, the latter is equivalent to (ν) (ν) checking whether Lχ (ym,χ ) satisfies the conditions (C1)–(C4) listed in Theorem 15. (ν) (ν) Theorem 30 If ∈ {III,IV,V,VI}, then Lχ (ym,χ ) satisfies conditions (C1)–(C4) in Theorem 15 if and only if: (ν) (ν) (i) N Stab (χ )/ pm ymχ ,χ = ym,χ at each m ∈ {mχ , . . . , n − s}, (ν)
(ν)
(ii) ym,χ = ym,χ whenever χ ∈ ∗ χ, and (iii) (ν) (ν) Tr( Indχ ∗ )( ) ϕ N0,mχ −1 y1 yχ N Stab (χ )/ pm (ν) · (ν) ϕ yχ p N0,mχ y1 χ ∈Rm,∞
(m) (m) m ν
mod p N,1 +N,2 +m− ord p (# ) · Z p p / p (m,∞) . for every integer m ∈ {0, . . . , ν}, and every orbit ∈ orb H∞
≡ 1
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K1 -congruences for three-dimensional
(m,ν) (ν) (ν) Proof If one chooses the sequence aχ := Lχ (ym,χ ) , then (C1) is readily seen to be equivalent to (i), while condition (C2) is equivalent to (ii). Focussing therefore on conditions (C3) and (C4), if one puts eχ∗ , = Tr Indχ ∗ ( ) then
Tr Stab (χ )/ pm aχ(ν) · Tr Indχ ∗ ( ) =
χ ∈Rm,n
by (19)
=
eχ∗ ,
χ ∈Rm,n
=
× Tr Stab (χ )/ pm ◦ log ⎛
log pm ⎝ Zp
p
ν
χ ∈Rm,n
χ ∈Rm,n (ν)
yχ
N0,mχ y1(ν)
N Stab (χ )/ pm
Recall that (C3) and (C4) together imply
(ν)
yχ
N0,mχ y1(ν)
(
(m,n)
eχ∗ , × Tr Stab (χ )/ pm aχ(ν) · ϕ pmχ −1 p
(ν)
ν
· ϕ pmχ −1 p
ν
χ ∈Rm,n Tr Stab (χ )/ p
m
N0,mχ −1 y1(ν) yχ p
eχ∗ ,⎞ N0,mχ −1 y1(ν) ⎠. (ν) yχ p
(ν) aχ · Tr Indχ ∗ ( ) is ν
congruent to zero modulo p ord p (#H∞ )+m− ord p (# ) ·Z p [ p / p ], for m ∈ {0, . . . , n −s} (m,n) and at each orbit ∈ orb H∞ . Now for all integers i ≥ 1, the mappings log : ∼
ν
m
m
ν
m
m
∼
ν
1 + pi · Z p [ p / p ] −→ pi · Z p [ p / p ] and exp : pi · Z p [ p / p ] −→ 1 + pi · m ν Z p [ p / p ] are inverse isomorphisms to each other. As an immediate consequence, χ ∈Rm,n
Tr Stab (χ)/
if and only if
pm
(ν) aχ · Tr Indχ ∗ ( ) ≡ 0
χ ∈Rm,n
(m,n)
N Stab (χ )/ pm
(ν)
yχ
(ν)
ϕ yχ p
mod p
(m,n)
ord p (#H∞
·
(ν)
ϕ N0,mχ −1 y1
(ν)
N0,mχ y1
4 )+m− ord p (# )
· Zp
m
p ν p
5
Tr( Indχ ∗ )( )
belongs to
m ν · Z p [ p / p ]. (m,n) ∼ (m,∞) Finally, both H∞ and Rm,n = Rm,∞ provided that ∈ {III,IV,V,VI}; = H∞ (m,n) (m) (m) = N,1 + N,2 , therefore the equivalence is fully established. moreover ord p #H∞
1 + p ord p (#H∞
)+m− ord p (# )
The reader will notice that these congruences are independent of the choice of n ≥ m + s. They also behave well if we take the projective limit as ν → ∞, hence one can obtain m
m ν analogous congruences for the completed group algebras Z p p = limν Z p [ p / p ], ← − i.e. those congruences labelled Eq. (2) in §1.2. The proof of the ‘non-S -localised version’ of Theorem 2 has therefore been completed, m × belongs to ∞,χ K 1 ( (G∞ )) if and only i.e. a sequence ym,χ ∈ m,χ OC p p (ν) (ν) (ν) (ν) if N Stab (χ )/ pm ymχ ,χ = ym,χ if m ≥ mχ , secondly ym,χ = ym,χ for χ ∈ ∗ χ, and lastly χ ∈Rm,∞
N Stab (χ )/ pm
Tr( Indχ ∗ )( ) ϕ N0,mχ −1 y1 yχ · ϕ yχ p N0,mχ y1
(m) (m) m
≡ 1 mod p N,1 +N,2 +m− ord p (# ) · Z p p
(m,∞) . for every positive integer m, and at every orbit ∈ orb H∞
123
D. Delbourgo, Q. Chao
Remarks (a) If =II, the proof of Theorem 1 runs along identical lines - the only point of (m) (m) departure is that N II,1 = n and N II,2 = s + m, so Rm,n is no longer independent of n. Nevertheless in Case (II), the multiplicative conditions equivalent to (C3) and (C4) are χ ∈Rm,n
N Stab (χ )/ pm
Tr( Indχ ∗ )( ) ϕ N0,mχ −1 y1 yχ · ϕ yχ p N0,mχ y1
m
≡ 1 mod p s+2m+n− ord p (# ) · Z p p
(20)
(m,n) . for every positive integer m ≤ n − s, and at every orbit ∈ orb H∞ (b) To transform these into the congruences labelled Eq. (1), one must calculate each of Rm,n , # and Tr( Indχ ∗ )( ) precisely – we refer the reader to the worked example given later in Sect. 5.1, for the full details. (c) Of course, this stillonly gives us a non-S -localised version of Theorem 1, describing ∞,χ K 1 ( (G∞ )) rather than ∞,S ,χ K 1 (G∞ )S , which is an issue we address below. Extending these congruences to the localisations. Finally, we these explainhow to extend results from K 1 (G∞ ) , to both of the Ore localisations K 1 (G∞ )S and K 1 (G∞ )S ∗ . Let us focus first on K 1 (G∞ )S , and write ∞,S : K 1 (G∞ )S → K 1 (Umab )S m≥0
for the corresponding collection of morphisms θm,S , with θm,S := NUm (−) mod [Um , Um ]. In order to extend the arguments in Sects. 4.1, 4.2 and 4.3 so as to produce non-abelian congruence conditions ‘S ’ describing Im ∞,S , one must first extend the Taylor-Oliver logarithm to a homomorphism ( G∞,n )S G∞,n )S −→ LOGG∞,n ,S : K 1 (
for every n ≥ 1, G∞,n )S (G∞,n )S , ( where ( G∞,n )S denotes the Jac Z p [H∞,n ] -adic completion of the localisation (G∞,n )S . This task has already been partially accomplished (see for example [7, Sect. 5] or [17]), but not enough is known about the kernel and cokernel of these maps on the completion. Indeed by [7, Lemma 5.2], the extension of the logarithm sits inside a commutative square K 1 (G∞,n ) −→ K 1 ( G∞,n )S ⏐ ⏐ LOG LOG G∞,n
Z p Conj(G∞,n ) −→
G∞,n ,S
G∞,n )S (
G∞,n )S , ( G∞,n )S (
where the horizontal arrows are induced from the natural inclusion (G∞,n ) → ( G∞,n )S . We simply observe that the properties of the Taylor-Oliver logarithm we derived in Sect. 4.3 extend to the Jac Z p [H∞,n ] -adic completion if one ignores their kernels/cokernels, and omit the details (which are anyway identical to Sect. 5 of op. cit.). The remainder of the proof of Theorems 1 and 2 in the S -localised situation then follows readily, albeit the congruences
123
K1 -congruences for three-dimensional
m
in Eqs. (1) and (2) are now taken modulo p • · Z p p ( p) rather than just modulo p • · m
Z p p , and we unfortunately lose their sufficiency in the process. We now turn our attention to the S ∗ -localisation, (G∞ )S ∗ , which is less problematic. Recall that G∞ has no element of order p, in which case Burns and Venjakob [3, Prop 3.4] have constructed a splitting K 1 (G∞ )S ∗ ∼ = K 1 (G∞ )S ⊕ K 0 F p [[G∞ ]] . Furthermore, there exists another commutative diagram ∼ K 1 (G∞ )S ∗ −→ K 1 (G∞ )S ⊕ K 0 F p [[G∞ ]] ⏐ ⏐ ∗ ( , ) 0 ∞,S ∞,S ab ∗ ab ← K 1 (Um )S K 1 (Um )S ⊕ K 0 F p [[Umab ]] m≥0
m≥0
K 0 F p [[Umab ]] encodes how the nonwhere the map 0 : K 0 F p [[G∞ ]] → m≥0 commutative μ-invariant information in K 0 F p [[G∞ ]] gets distributed amongst its abelian fragments. Thus a sequence (yS ∗,m ) lies in the image of ∞,S ∗ , if and only if each term factorises into yS ∗,m = yS ,m , μm where the components (yS ,m ) ∈ Im ∞,S and (μm ) ∈ Im(0 ). Note that G∞ is a pro- p-group so that K 0 F p [[G∞ ]] ∼ = Z, and similarly K 0 F p [[Umab ]] ∼ = Z. Consequently a tuple (μm ) ∈ m K 0 F p [[Umab ]] arises from the image of 0 if and only if for every integer m ≥ 0, one has μm = [G∞ : Um ] × μ for some fixed μ ∈ Z. Because the bottom arrow in the above diagram may possibly not be surjective, the most one can say is that any (yS ∗,m ) ∈ Im ∞,S ∗ must of necessity satisfy (M1)–(M4). If we denote this subset of m≥0 K 1 (Umab )S ∗ satisfying (M1)–(M4) by ‘S ∗ ’, then this potential lack of surjectivity yields another obstruction to ∞,S ∗ : K 1 (G∞ )S ∗ → S ∗ being an isomorphism. In terms of ∞,χ ,S ∗ = χ ◦ ∞,S ∗ from the Introduction, this translates into the necessity of the congruences written down in Theorems 1 and 2 holding m × for χ (yS ∗,m ) ∈ m,χ Quot Oχ ( p ) , but not their sufficiency regrettably.
5 Computing the terms in Theorems 1 and 2 The various quantities Rm,n , and eχ∗ , occurring in the congruences (1) and (2) are easy to define in theory, but it is not quite so evident how to work them out in practice. We shall now give a step-by-step guide to calculating these terms algorithmically. Step 1: We first explain how to express χ˜ 1,N (m) and χ˜ 2,N (m) in terms of χ1,n and χ2,n . ,1
,2
Step 2: We next explicitly list representatives for Rm,n in the form χ˜ a Step 3: We end by giving formulae to compute both # and eχ∗ ,
· χ˜ b
(m) . 2,N,2 = Tr Indχ ∗ ( ). (m)
1,N,1
The technical results corresponding to Steps 1, 2, 3 in the text below are respectively Proposition 32, Lemmas 34 and 35.
123
D. Delbourgo, Q. Chao
[1,m] [1,m] equal to Definition 31 (a) We set the non-negative integer pair e,1 , e,2 ! ! " ! "" "! p s+m p s+m p s+m 1 1 1 1 √ • 0, 1 , , 0 + pm − pm pm pm pm 2 2 d λ III,± −1 λ IV,+ −1 λ IV,− −1 λ IV,+ −1 λ IV,− −1
r r ! " p p • ! •
1− √
p s+m+ ord p (d) 2
p s+m 2
!
2 V pm λV,+ −1
pr+ ord p (t) pm λVI,+ −1
1+ √
2 V pm
+
λV,− −1
pr+ ord p (t) pm λVI,− −1
+
"
,
,
ord p (d) p s+m+ √ 2 V
p√s+m 2 pr t
!
1 pm λV,+ −1
pr+ ord p (t) pm λVI,+ −1
−
−
pr+ ord p (t) pm λVI,− −1
1 pm λV,− −1
""
in Cases (II), (III), (IV), (V) and (VI) respectively. [2,m] [2,m] (b) Likewise, we shall define a second pair e,1 , e,2 by setting it equal to • 1, 0 " ! p s+m • 0 , pm λ III,± −1 " ! "" ! √ ! s+m p d p s+m 1 1 1 1 − pm + pm , 2 • pm pm 2 λ IV,+ −1 λ IV,− −1 λ IV,+ −1 λ IV,− −1
r ! " p pr • ! •
s+m d p√ 2 V
1
pm λV,+ −1
√ p s+m pr t 2
!
−
1
pm
,
1
pm λV,− −1
λVI,+ −1
−
" 1
pm
λVI,− −1
1+ √
p s+m 2
,
2 V pm λV,+ −1
p s+m 2
+
!
1
pm
λVI,+ −1
1− √
2 V pm
λV,− −1
+
""
1
pm
λVI,− −1
again in Cases (II), (III), (IV), (V) and (VI) respectively. Proposition 32 For integers n 0, one has the character relations ⎧ 0 1 χ1,n · χ2,n if = II ⎪ ⎪ ⎪ ⎪ ⎪ [1,m] ⎪ e III,1 ⎪ 0 ⎪ χ1,s+m · χ2,s+m if = I I I ⎪ ⎪ ⎪ ⎪ [1,m] ⎨ e[1,m] e IV,2 IV,1 · χ2,s+m if = IV χ1,s+m χ˜ 1,N (m) = ⎪ ,1 ⎪ [1,m] [1,m] ⎪ e ⎪ eV,1 ⎪ ⎪ χ · χ V,2 if = V ⎪ ⎪ 1,s+m+ ord p (d) 2,s+m+ ord p (d) ⎪ ⎪ [1,m] ⎪ ⎪ e[1,m] ⎩χ eVI,1 VI,2 1,s+m+r +ord p (t) · χ2,s+m+r + ord p (t) if = VI and
χ˜ 2,N (m) ,2
⎧ 1 ⎪ χ1,s+m ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪χ1,s+m ⎪ ⎪ ⎪ ⎨ e[2,m] IV,1 = χ1,s+m ⎪ ⎪ ⎪ ⎪ e[2,m] ⎪ V,1 ⎪ χ ⎪ 1,s+m ⎪ ⎪ ⎪ [2,m] ⎪ ⎪ ⎩χ eVI,1 1,s+m
0 · χ2,s+m if = II e[2,m]
III,2 · χ2,s+m if = III
e[2,m]
IV,2 · χ2,s+m if = IV
e[2,m]
V,2 · χ2,s+m if = V
e[2,m]
VI,2 · χ2,s+m if = VI.
Proof The situation where =II has already been dealt with in Sect. 3.2, cf. Eq. (13). Let us instead suppose ∈ {III,IV,V,VI}. We first recall from Definition 16 that
123
K1 -congruences for three-dimensional
!" !! x 1 = χ1,N (m) • χ˜ 1,N (m) 0 ,1 y ,1 !" !! x 0 = χ2,N (m) • χ˜ 2,N (m) 0 ,2 y ,2 (m)
where T,m, j := p N, j
I2 + M
pm
" ! "" 0 x ,and T,m,1 0 y " ! "" 0 x T,m,2 1 y
I2 + M
pm
pm
= P D P−1
− I2
−1
. Further, one can diagonalise the γ -action via " ! λ,+ 0 and P ∈ GL2 (Q p ). 0 λ,−
withD =
The next objective is to calculate the matrices T,m, j on an individual, case-by-case basis. (m) (m) Case (III). Here PIII = I2 and N III,1 = N III,2 = s + m, so that p
(m)
N III, j
⎞
⎛ p s+m −1 m m p (1+ p s ) p −1 ⎝ (I2 + M) − I2 = 0
0 p s+m m (1+ p s ) p −1
⎠.
! " 1 (m) (m) √1 √ and N I V,1 = N I V,2 = s + m, so that for each d− d −1 (m) m equals j ∈ {1, 2}, the matrix p N I V, j (I2 + M) p − I2
Case (IV). Here PI V =
⎛ p s+m 2
"⎞
!
⎜ ⎜ ! ⎜ ⎝√ d
1
pm λ IV,+ −1
+
1 pm λ IV,+ −1
1
pm λ IV,− −1
−
"
1 pm λ IV,− −1
−
1
√1 d
pm λ IV,+ −1
+
1 pm λ IV,+ −1
1
pm λ IV,− −1
1 pm λ IV,− −1
⎟ ⎟ ⎟. ⎠
" 1 1 √ √ r with p 2 + V 2 − V (m) (m) V = d + p 2r /4 ∈ Z p , while N V,1 = s + m + ord p (d) and N V,2 = s + m; consequently −1 (m) m for each choice j ∈ {1, 2}, the matrix p NV, j (I2 + M) p − I2 equals !
Case (V). Assume that n ≥ s + m + ord p (d). Then PV =
⎛ (m)
p NV, j 2
r √p 2 V
⎜λ pm −1 + λ pm −1 − ⎜ V,+ V,− ! ⎜ ⎝ d 1 √ − pm1 pm 1
1
V
λV,+ −1
"
!
pr
" 1
pm
λV,+ −1
−
1 pm λV,+ −1
λV,− −1
√1 V
pm
λV,− −1
+
1 pm λV,− −1
"⎞
!
1
r √p 2 V
+
1
pm
!
λV,+ −1 1 pm λV,+ −1
− −
1
pm
λV,− −1 1 pm λV,− −1
⎟ ⎟ "⎟ . ⎠
" ! 1 √1 r , Case (VI). Assume that n ≥ s + m + r + ord p (t). Then one has PVI = √ r p t − p t (m) (m) while N VI,1 = s + m + r + ord p (t) and N VI,2 = s + m; consequently, for each j ∈ {1, 2} −1 (m) m the matrix p NVI, j (I2 + M) p − I2 equals ⎛ p
(m) N VI, j
2
⎜ ⎜ ⎜ ⎝√
"⎞
! 1
pm
λVI,+ −1
+
!
pr t
1 pm λVI,+ −1
1
pm
λVI,− −1
−
1 pm λVI,− −1
"
√1 r p t
1
pm
λVI,+ −1 1 pm λVI,+ −1
+
−
1
pm
λVI,− −1
1 pm λVI,− −1
⎟ ⎟ ⎟. ⎠
123
D. Delbourgo, Q. Chao
!" !" x x and χ˜ 2,N (m) . To Since we know the form of each T,m, j , one now computes χ˜ 1,N (m) ,1 y ,2 y illustrate the calculation, suppose we are in the last case =VI; then one obtains !" !! " ! "" x 10 x = χ1,N (m) χ˜ 1,N (m) TVI,m,1 00 y VI,1 y VI,1
⎞⎞ ⎛⎛ y y ⎜⎜ = χ1,s+m+r + ord p (t) ⎝⎝ ⎛⎛ = χ1,s+m+r + ord p (t)
x+ √
p s+m+r+ ord p (t) 2
p s+m ⎝⎝ 2
!
⎛⎛ · χ2,s+m+r + ord p (t) ⎝⎝ p√s+m
2 pr t
pr t pm λVI,+ −1
+
x− √
0 pr+ ord p (t) pm λVI,+ −1
+
pr t
pm
λVI,− −1
pr+ ord p (t) pm λVI,− −1
" ⎞⎞ x⎠⎠
0 !
0 pr+ ord p (t) pm λVI,+ −1
−
⎟⎟ ⎠⎠
pr+ ord p (t) pm λVI,− −1
"
⎞⎞ ⎠⎠ y
!" !" e[1,m] x x VI,2 which equals · χ2,s+m+r + ord p (t) . Likewise, one can show that y y !" !! " ! "" x 00 x TVI,m,2 χ˜ 2,N (m) = χ2,s+m 01 y VI,2 y " ⎞⎞ ⎛⎛ √ r ! s+m p p t 1 − pm1 x⎠⎠ pm 2 = χ1,s+m ⎝⎝ λVI,+ −1 λVI,− −1 0 ⎞⎞ ⎛⎛ !" !" 0 ! " e[2,m] e[2,m] x x VI,1 VI,2 ⎠ ⎠ = χ · χ · χ2,s+m ⎝⎝ ps+m 1 1,s+m y 2,s+m y . y + pm1 pm 2 e[1,m] VI,1 χ1,s+m+r + ord p (t)
λVI,+ −1
λVI,− −1
The other remaining cases =III, =IV and =V follow in an analogous fashion.
For Step 2, we introduce an equivalence relation ‘ ∼ ’ on ordered pairs of integers (a, b). Definition 33 (i) If ∈ {III, IV, V, VI }, then one sets
: ; Z Z Z Z Xm,n := (a, b) ∈ × × ∼ − p· (m) (m) (m) (m) p N,1 Z p N,2 Z p N,1 Z p N,2 Z where (a, b) ∼ (a , b ), if and only if ! " ! " j pm a0 a 0 I2 + M mod I2 + M − I2 for some j ∈ Z/ p m Z. ≡
0 b 0b (ii) If = II , then one sets Xm,n :=
Z (a, b) ∈ n × p Z
!
Z
"× : ;
p s+m Z
∼
where (a, b) ∼ (a , b ) if and only if a ≡ a ( mod p n−m ). The following result describes how to produce an explicit set of representatives for Rm,n . Again we assume that the integer n 0 is chosen sufficiently large with respect to m.
123
K1 -congruences for three-dimensional
Lemma 34 (a) Up to isomorphism, the exact number of irreducible G∞,n -representations (m,n) G → C× ρχ = Ind ∞,n (m,n) (χ) induced from primitive characters χ : H∞ Stab (χ )H∞
equals
#Rm,n − #Rm−1,n
⎧ n+s−1 p × ( p − 1) ⎪ ⎪ ⎪ ⎨ p 2s+m−2 × ( p 2 − 1) = ⎪ p 2s+m+ ord p (d)−2 × ( p 2 − 1) ⎪ ⎪ ⎩ 2s+m+r + ord p (t)−2 p × ( p 2 − 1)
in Case (II) in Cases (III) and (IV) in Case (V) in Case (VI).
prim
(b) If we define Rm,n := Rm,n − Rm−1,n . . . , n − s}, then we can take for every m ∈ {1, prim a b as representatives for Rm,n the set χ˜ (m) · χ˜ (m) (a, b) ∈ Xm,n . 1,N,1
2,N,2
Proof Part (a) follows (with n m) on combining Proposition 10(iii) 11. To and Corollary
a b a b j show (b), first suppose that = II. Then χ˜ (m) · χ˜ (m) = γ ∗ χ˜ (m) · χ˜ (m) if and 1,N,1 2,N,2 1,N,1 2,N,2 ! " ! " ax bx only if χ˜ 1,N (m) · χ˜ 2,N (m) equals ay by ,1 ,2 ! ! "" ! ! "" ax j b x · χ ˜ for all x, y ∈ Z p . (I + M) χ˜ 1,N (m) (I2 + M) j (m) 2 2,N,2 a y b y ,1 This latter equality is equivalent to the pair of congruences (m) −1 !ax " pm p N,1 0 I2 + M − I2 ay 0 0 (m) ! " −1 pm ax p N,1 0 I2 + M − I2 (I2 + M) j ≡ a y 0 0
(m)
mod p N,1
and
≡
0
0
(m) N,2
0 p 0
0 p
0
(m) N,2
I2 + M
pm
I2 + M
− I2 pm
−1 !bx "
− I2
by −1
(I2 + M) j
! " bx b y
(m)
mod p N,2
holding for all x, y ∈ Z p ; here we have exploited the construction of χ˜ 1,N (m) and χ˜ 2,N (m) ,1 ,2 pm given in Definition 16. Because I2 + M − I2 and (I2 + M) j commute with each other, the above may be rewritten as a single congruence ! " −1 pm a 0 I2 + M − I2 0b ! " −1 pm a 0 ≡ − I2 mod Mat2×2 Z p . (I2 + M) j I2 + M
0 b Note this congruence is satisfied for some j ∈ Z/ p m Z precisely when (a, b) ∼ (a , b ).
123
D. Delbourgo, Q. Chao
Let us instead suppose that = II. Then χ˜ a
(m)
1,N,1
· χ˜ b
(m)
2,N,2
= γ j ∗ χ˜ a
(m)
1,N,1
· χ˜ b
(m)
2,N,2
if
and only if " " ! " ! " !
!
ax bx a (x + p s j y) b (x + p s j y) χ˜ 1,N (m) · χ˜ 2,N (m) · χ˜ 2,N (m) = χ˜ 1,N (m) a y b y ay by ,1 ,2 ,1 ,2 at every x, y ∈ Z p . Again using Definition 16, we can rewrite this as a y
ay
b (x+ p s j y)
ζ pn · ζ pbxs+m = ζ pn · ζ ps+m
for eachx, y ∈ Z p ,
which is itself equivalent to the congruences b ≡ b ( mod p s+m )
a ≡ a + j p n−m b ( mod p n ) for some j ∈ Z/ p m Z.
and
These last two congruences then reduce to b ≡ b ( mod p s+m ) and a ≡ a ( mod p n−m ). Therefore in all possible cases ∈ {II,III,IV,V,VI}, one concludes that χ˜ a (m) · χ˜ b (m) and χ˜ a
(m)
1,N,1
· χ˜ b
(m)
2,N,2
lie in the same -orbit if and only if (a, b) ∼ (a , b ).
1,N,1
2,N,2
Consequently Steps 1 and 2 have now been resolved, and it therefore only remains to complete Step 3. The latter task is covered by the next result, which enables us to compute both the size of and also the exponent eχ∗ , occurring in Theorems 1 and 2, for each orbit and representative character χ ∈ Rm,n . (m,n) x y Lemma 35 (i) If ∈ orb H∞ contains an element h = h 1 h 2 , then $ =
a b
h 1 h 2 such that
! " !Z " ! p n Z "% pm a p ∈ Y(x,y) mod I2 + M − I2 + n p b Zp p Zp
! " % j x m with j = 0, 1, . . . , p − 1 . where the set Y(x,y) consists of the vectors I2 + M y (m,n) (ii) For each character χ = χ˜ a (m) ·χ˜ b (m) on H∞ , the number eχ∗ , = Tr Indχ ∗ ( ) $
1,N,1
2,N,2
can be computed via the exponential sum formula p
mχ −m
·
m −1 p
√
exp −2π −1
[1,m] ae,1
j=0
(m)
p N,1
+
[2,m] be,1
(m)
p N,2
xj +
[1,m] ae,2 (m)
p N,1
+
[2,m] be,2 (m)
p N,2
yj
! " ! " j x xj := I2 + M for all j. yj y (iii) In particular, if consists of just the identity element, then eχ∗ , = p mχ ∈ N. where the integer mχ is given in Proposition 5, and
Proof To establish assertion (i), we remark that γ acts on the quotient group (m,n) H∞
pn
= &
H∞ /H∞ y h 1x h 2
mod
pn H∞ ,
γ
pm
' ∼ =
x, y ∈ Z p
Z p
(m) N,1
Z
×
Z (m)
p N,2 Z
through the matrix I2 + M, hence our description for the -orbit follows immediately.
123
K1 -congruences for three-dimensional
To show part (ii), by the definition of Tr Indχ ∗ ( ) one calculates that eχ∗ ,
= by 5
=
p −1 p −1 [ : Stab (χ)] −1 x j y j #( ∗ χ) −1 − j j γ = · χ hγ χ h1 h2 · m pm [ : p ]
p mχ −m ·
m
m
j=0
j=0
m −1 p
x y −a x y −b χ˜ 1,N (m) h 1 j h 2 j × χ˜ 2,N (m) h 1 j h 2 j ,1
j=0 by 32
=
p
mχ −m
·
m −1 p
χ
(m) 1,N,1
j=0
=
p mχ −m ·
p m −1
j=0
[1,m] e,1
χ
,2
·χ
(m) 2,N,1
[1,m] −ae,1
(m) 1,N,1
[1,m] e,2
·χ
[2,m] −be,1
(m) 1,N,2
x
y
h1 j h2 j
−a
×χ
[2,m] e,1
(m) 1,N,2
·χ
[2,m] e,2
(m) 2,N,2
x
y
h1 j h2 j
−b
[1,m] [2,m] −ae,2 −be,2 x y x y h 1 j h 2 j × χ (m) · χ (m) h1 j h2 j 2,N,1
2,N,2
and the last line is then equivalent to the stated formula. Finally (iii) is a special case of (ii), corresponding to x = y = 0 and x j = y j = 0.
5.1 A worked example for Case (II) We end by using Steps 1–3 to yield an explicit expression for the congruences in Case (II). Firstly by Lemma 34(b) and Definition 33(ii), if one takes m ≥ 1 then × prim a b Rm,n = χ2,n · χ1,s+m a ∈ Z/ p n−m Z and b ∈ Z/ p s+m Z a · χ b a ∈ Z/ p n Z and b ∈ Z/ p s Z . It follows that while R0,n coincides with χ2,n 1,s
∗ eχ,
N Stab (χ )/ pm (· · · )
χ ∈Rm,n
=
m m =0
n−m p s+m p
a=1 b = 1,
Nmχ ,m (· · · )
p b if m > 0
∗ eχ,
. a ·χ b χ =χ2,n 1,s+m
(m,n) x y Now suppose an orbit h ∈ orb H∞ contains an element h = h 1 h 2 . Then x+ j ps y y h = γ − j hγ j j ∈ Z = h 1 h2 j ∈ Z j ps y = h · h1 j = 1, · · · , p m− ord p (y) in which case #h = p m− ord p ( y˜ ) , with y˜ ∈ {1, . . . , p m } chosen so that y˜ ≡ y ( mod p m ).
p m−m b
a · χb a Finally, if we consider a typical character χ = χ2,n 1,s+m = χ2,n · χ1,s+m and the orbit = h as above, then Lemma 35(ii) implies
m −1 ! "
p √ p m−m b a ∗ mχ −m s y · exp −2π −1 eχ , = p (x + j p y) + h p s+m pn j=0
= p
mχ −m
= p mχ −m
m −1 "" p "" ! ! ! ! √ √ bx ay bj y × · exp −2π −1 + exp −2π −1
pn p s+m pm j=0 ! ! ""
√ bx ay p m if p m | by · exp −2π −1 ×
+ pn p s+m 0 if p m by.
123
D. Delbourgo, Q. Chao
√ However the exponential term exp −2π −1
bx
s+m
p e1 χ1,n
+
ay pn
is then just equal to χ −1 h .
e2 a · χb · χ2,n with e1 = p n−s−m b and e2 = a, Because χ = χ2,n 1,s+m can be written as ˜ χ } where one calculates via Proposition 5 that mχ = max{0, m
by5 ˜ χ = n − s − ord p p n−s−m b = m − ord p (b). m
χ −1 (h) · p max{0,m − ord p (b)} if p m | by a b ∗ Consequently, if χ = χ2,n ·χ1,s+m then eχ , =
h 0 if p m by. Corollary 36 The congruences described in Eq. (20) are equivalent to ∗ n−m
s+m
eχ, p p m h ϕ N0,mχ −1 y1 yχ · Nmχ ,m ϕ yχ p N0,mχ y1
b = 1, a=1 m =0
≡ 1
p b if m > 0
a ·χ b χ =χ2,n 1,s+m
m
mod p s+m+n+ ord p ( y˜ ) · Z p p ( p) x˜ y˜
(m,n)
for all integer pairs m, n ≥ 0 with m ≤ n − s, and at every choice of h = h 1 h 2 ∈ H∞ with x˜ ∈ {1, . . . , p n } and y˜ ∈ {1, . . . , p m }. This completes the proof of Theorem 1, in the precise form stated in the Introduction.
Acknowledgements The authors are grateful to both Antonio Lei and Lloyd Peters for numerous discussions about non-commutative congruences. They were also hugely inspired by the work of Mahesh Kakde, to which many arguments in this paper owe a great debt. Lastly they thank Ian Hawthorn for his friendly guidance during some difficult times.
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