Regulator constants and the parity conjecture
aa r X i v : . [ m a t h . N T ] M a r . REGULATOR CONSTANTS AND THE PARITYCONJECTURE
TIM † AND VLADIMIR DOKCHITSER
Abstract.
The p -parity conjecture for twists of elliptic curves relatesmultiplicities of Artin representations in p ∞ -Selmer groups to root num-bers. In this paper we prove this conjecture for a class of such twists. Forexample, if E/ Q is semistable at 2 and 3, K/ Q is abelian and K ∞ is itsmaximal pro- p extension, then the p -parity conjecture holds for twists of E by all orthogonal Artin representations of Gal( K ∞ / Q ). We also giveanalogous results when K/ Q is non-abelian, the base field is not Q and E is replaced by an abelian variety. The heart of the paper is a studyof relations between permutation representations of finite groups, their“regulator constants”, and compatibility between local root numbersand local Tamagawa numbers of abelian varieties in such relations. Contents
1. Introduction 21.i. Parity conjectures 21.ii. G -sets versus G -representations 31.iii. Main results and applications 41.iv. Regulator constants and parity of Selmer ranks 71.v. Root numbers and Tamagawa numbers 91.vi. Notation 102. Functions on the Burnside ring 112.i. Relations between permutation representations 112.ii. Regulator constants 142.iii. Functions modulo G -relations 182.iv. D ρ and T Θ ,p Date : March 21, 2009.
MSC 2000:
Primary 11G05; Secondary 11G07, 11G10, 11G40, 19A22, 20B99. † Supported by a Royal Society University Research Fellowship. Introduction
The emphasis of this paper is twofold: to study the interplay betweenfunctions on G -sets and on G -representations for a finite group G , and touse it to link root numbers and Tamagawa numbers of abelian varieties.The main application is the parity conjecture for classes of twists of ellipticcurves and abelian varieties by Artin representations.1.i. Parity conjectures.
Consider an abelian variety A defined over a num-ber field K , and a Galois extension F/K . The Galois group Gal(
F/K ) acts onthe F -rational points of A , and an extension of the Birch–Swinnerton-Dyerconjecture relates the multiplicities of complex representations in A ( F ) ⊗ C to the order of vanishing of the corresponding twisted L -functions at s = 1: Conjecture 1.1 (Birch–Swinnerton-Dyer–Deligne–Gross; [40, 4], [28] § . For every complex representation τ of Gal(
F/K ) , h τ, A ( F ) i = ord s =1 L ( A, τ, s ) . (Here and below h τ, ∗ i is the usual representation-theoretic inner productof τ and the complexification of ∗ .) When τ is self-dual, the parity of theright-hand side is forced by the sign in the conjectural functional equation,the global root number w ( A/K, τ ). Thus, we expect( − h τ,A ( F ) i = w ( A/K, τ ) . There is an analogous picture for Selmer groups. For a prime p , let X p ( A/K ) = (
Pontryagin dual of the p ∞ -Selmer group of A/K ) ⊗ Q p . This is a Q p -vector space whose dimension is simply the Mordell-Weil rankof A/K plus the number of copies of Q p / Z p in the Tate-Shafarevich group X ( A/K ). The conjectural finiteness of X then suggests the following paritystatements, which are often much more accessible: Conjecture 1.2a ( p -Parity Conjecture) . ( − dim X p ( A/K ) = w ( A/K ) . Similarly, for a self-dual representation τ of Gal( F/K ), we expect
Conjecture 1.2b ( p -Parity Conjecture for twists) . ( − h τ, X p ( A/F ) i = w ( A/K, τ ) . When K ⊂ L ⊂ F , the second statement for the permutation representationon the set of K -embeddings L ֒ → F is equivalent to the first one for A/L . So1.2b for all orthogonal twists implies 1.2a over all intermediate fields of
F/K .The main applications of this paper confirm special cases of Conjec-ture 1.2b. Here are two specific examples:
Theorem 1.3.
Let E/ Q be an elliptic curve, semistable at 2 and 3. Sup-pose F/ Q is Galois and the commutator subgroup of G = Gal( F/ Q ) is a p -group. Then the p -parity conjecture holds for twists of E by all orthogonalrepresentations of G and, in particular, over all subfields of F . EGULATOR CONSTANTS AND THE PARITY CONJECTURE 3
Theorem 1.4.
Let p be an odd prime, and suppose F/K is Galois and
P ⊳
Gal(
F/K ) is a p -subgroup. Let A/K be a principally polarised abelianvariety whose primes of unstable reduction are unramified in
F/K . If the p -parity conjecture holds for A over the subfields of F P /K , then it holdsover all subfields of F/K . The general results on the p -parity conjecture (Theorems 1.6, 1.11 and 1.12)are given in § G -sets versus G -representations. Let G be an abstract finite group.Suppose φ : H φ ( H ) is a function that associates to every subgroup H < G a value in some abelian group A (written multiplicatively), and that φ takes the same value on conjugate subgroups. Recall that H ↔ G/H isa bijection between subgroups of G up to conjugacy and transitive G -setsup to isomorphism. So φ extends to a map from all G -sets to A by the rule φ ( X ∐ Y ) = φ ( X ) φ ( Y ). Let us call φ “representation-theoretic” if φ ( X ) onlydepends on the representation C [ X ].Alternatively, say that a formal combination of (conjugacy classes of)subgroups Θ = P i n i H i is a relation between permutation representationsof G , or simply a G -relation , if M i C [ G/H i ] ⊕ n i ∼ = 0 , as a virtual representation, i.e. the character P i n i χ C [ G/Hi ] is zero. Thenfor φ to be representation-theoretic is equivalent to Q i φ ( H i ) n i being 1 forevery such G -relation.For example, G = S has a unique relation up to multiples,Θ = 2S + { } − − C . (i.e. ⊕ ⊕ C [S ] ∼ = C [S / C ] ⊕ ⊕ C [S / C ]; clearly such a relation must exist:S has 4 subgroups up to conjugacy, but only 3 irreducible representations.)In the context of number theory, G may be a Galois group of a numberfield F/ Q , and φ ( H ) some invariant of the intermediate field F H . For in-stance, φ ( H ) could be the the degree of F H , its discriminant, class numberor Dedekind zeta-function ζ F H ( s ) (with A = Z , Q × , Q × and the group ofnon-zero meromorphic functions on C , respectively). Of these four, all butthe class number are representation-theoretic, e.g. [ F H : Q ] = dim C [ G/H ]and ζ F H ( s ) = L ( C [ G/H ] , s ) are visibly functions of C [ G/H ]. It follows, forexample, that in every S -extension F/ Q , ζ ( s ) ζ F ( s ) = ζ F C2 ( s ) ζ F C3 ( s ) . The class number formula then yields an explicit identity between the cor-responding class numbers and regulators ( h · Reg | µ | is representation-theoretic).We are going to study extensively G -relations and functions on G -relations,and present techniques for verifying when a function or a quotient of twosuch functions is representation-theoretic (see § TIM AND VLADIMIR DOKCHITSER
Regulator constants.
Of particular interest to us is the function D ρ : H det( | H | h , i| ρ H ) ∈ K × / K × that, for a fixed self-dual K G -representation ρ ( K a field) with a G -invariantpairing h , i , computes the determinant of the matrix representing | H | h , i onany basis of the H -invariants ρ H . Its significance will become clear when wediscuss functions coming from abelian varieties. The fundamental propertyof D ρ is that if hh , ii is another pairing on ρ , then D h , i ρ / D hh , ii ρ is representation-theoretic. In other words, for every G -relation Θ = P i n i H i , the quantity C Θ ( ρ ) = Y i D ρ ( H i ) n i ∈ K × / K × is independent of the pairing. Following [6] we call C Θ ( ρ ) the regulatorconstant of ρ . (Their properties are discussed in § § Example 1.5.
Suppose G = D p n is dihedral with p = 2, and K = Q or Q p .The smallest subgroups { } , C , C p and D p form a G -relationΘ = { } + 2 D p − C p − . The irreducible K G -representations are (trivial), ǫ (sign) and ρ k of dimen-sion p k − p k − for every 1 ≤ k ≤ n ; they are all self-dual. An elementarycomputation (see Examples 2.20, 2.21) shows that C Θ ( ) = C Θ ( ǫ ) = C Θ ( ρ n ) = p, C Θ ( ρ k ) = 1 (1 ≤ k < n ) . Main results and applications.
The central result of this paper isthe p -parity conjecture for the following twists: for a group G , a prime p anda G -relation Θ, define T Θ ,p to be the set of self-dual ¯ Q p G -representations τ that satisfy h τ, ρ i ≡ ord p C Θ ( ρ ) mod 2for every self-dual Q p G -representation ρ (computing C Θ ( ρ ) with K = Q p ). Theorem 1.6(a).
Let
F/K be a Galois extension of number fields. Suppose
E/K is an elliptic curve whose primes of additive reduction above 2 and 3have cyclic decomposition groups (e.g. are unramified) in
F/K . For every p and every relation Θ between permutation representations of Gal(
F/K ) , ( − h τ, X p ( E/F ) i = w ( E/K, τ ) for all τ ∈ T Θ ,p . Theorem 1.6(b).
Let
F/K be a Galois extension of number fields. Sup-pose
A/K is a principally polarised abelian variety whose primes of unstablereduction have cyclic decomposition groups in
F/K . Let p be a prime, andassume that either • p = 2 , or • p = 2 , the principal polarisation is induced by a K -rational divisor,and A has split semistable reduction at primes v | of K which havenon-cyclic wild inertia group in F/K .For every relation Θ between permutation representations of Gal(
F/K ) , ( − h τ, X p ( A/F ) i = w ( A/K, τ ) for all τ ∈ T Θ ,p . EGULATOR CONSTANTS AND THE PARITY CONJECTURE 5
Remark 1.7.
In particular, if p is odd, Conjecture 1.2b holds for τ ∈ T Θ ,p for all semistable principally polarised abelian varieties over K . Remark 1.8.
In general, the representations in T Θ ,p simply encode theregulator constants. For instance, if { ρ j } are the irreducible self-dual Q p G -representations with ord p C Θ ( ρ j ) odd, then M j (any ¯ Q p -irreducible constituent of ρ j ) ∈ T Θ ,p . (This representation is automatically self-dual by Corollary 2.25). A generalelement in T Θ ,p differs from this one by an element of T ,p , a representationfor which the p -parity conjecture ought to “trivially” hold; it would be veryinteresting to have an intrinsic description of S Θ T Θ ,p , cf. Remarks 2.51, 2.58. Example 1.9 (Gal(
F/K ) = D p n , p odd) . Continuing Example 1.5, forevery ¯ Q p -irreducible (2-dimensional) constituent τ n of ρ n , ⊕ ǫ ⊕ τ n ∈ T Θ ,p . If A/K is an abelian variety that satisfies the assumptions of Theorem 1.6,e.g. A is semistable at primes that ramify in F/K , the p -parity conjectureholds for the twist of A by ⊕ ǫ ⊕ τ n . Applying this construction to theD p k -quotients of D p n , we deduce the p -parity conjecture for the twists of A by ⊕ ǫ ⊕ τ for every 2-dimensional irreducible representation τ of Gal( F/K ).As the p -parity conjecture is known to hold for elliptic curves over Q , andtherefore for their quadratic twists as well, we find Corollary 1.10 (Parity conjecture in anticyclotomic towers) . Let E/ Q bean elliptic curve, L an imaginary quadratic field and p an odd prime. If p = 3 ,assume that either E is semistable at or that splits in L . Then for everylayer L n of the Z p -anticyclotomic extension of L and every representation τ of Gal( L n / Q ) , ord s =1 L ( E, τ, s ) ≡ h τ, X p ( E/L n ) i mod 2 . In § p -subgroup. Based on Theorem 1.6, and using purely group-theoretic ma-nipulations we obtain (see Theorems 4.5, 4.2) Theorem 1.11.
Suppose
F/K is a Galois extension of number fields andthe commutator subgroup of G = Gal( F/K ) is a p -group. Let A/K be anabelian variety satisfying the hypotheses of Theorem 1.6. If the p -parityconjecture holds for A over K and its quadratic extensions in F , then itholds for all twists of A by orthogonal representations of G . When A = E is an elliptic curve and K = Q , the assumption on the p -parityconjecture is always satisfied, as we remarked above (in particular, we getTheorem 1.3). It also holds for those E/K that admit a rational p -isogenyunder mild restrictions on E at primes above p ; see Remark 4.6 for precisestatements, an extension to abelian varieties and a list of references. TIM AND VLADIMIR DOKCHITSER
The condition that the commutator of G is a p -group is equivalent to theSylow p -subgroup being normal with an abelian quotient. In other words, F should be a p -extension of an abelian extension of the ground field. Forinstance, the theorem applies when • G is abelian (any p ). • G ∼ = D p n is dihedral. • G is a 2-group and p = 2. • G is an extension of C by a p -group. • G ∼ = ( Z /p n Z ) ⋊ ( Z /p n Z ) × , for instance F = Q ( µ p n , pn √ m ), K = Q . • G ⊂ (cid:0) ∗ p ∗ ∗∗ (cid:1) in GL ( Z /p n Z ), for instance F = K ( C [ p n ]) for someelliptic curve C/K that admits a rational p -isogeny.Root numbers and parities of Selmer ranks in the last 3 cases have recentlybeen studied by Mazur–Rubin [20, 21], Hachimori–Venjakob [15] and one ofus (V.) [8], Rohrlich [31] and Coates–Fukaya–Kato–Sujatha [3]; see alsoGreenberg’s preprint [12]. This kind of extensions arise in non-commutativeIwasawa theory, where one has a tower F ∞ = S F n with Gal( F ∞ /K ) a p -adic Lie group. The Gal( F n /K ) all have a “large” normal p -subgroupwith a fixed “small” quotient. When this quotient is non-abelian , we have aweaker version of Theorem 1.11 (Theorems 4.3, 4.2; cf. also 4.4 for p = 2): Theorem 1.12.
Suppose
F/K is Galois and
P ⊳
Gal(
F/K ) is a p -subgroupwith p = 2 . Let E/K be an elliptic curve (resp. principally polarised abelianvariety) whose primes of additive reduction above 2, 3 (resp. all primes ofunstable reduction) have cyclic decomposition groups in
F/K . If the p -parityconjecture holds for E over the subfields of F P /K , then it holds over allsubfields of F/K . Example 1.13.
Let E/ Q be an elliptic curve, semistable at 2 and 3. Take p = 2 and F n = Q ( E [ p n ]), so Gal( F n / Q ) < GL ( Z /p n Z ). If the p -parityconjecture holds over the subfields of the first layer Q ( E [ p ]) / Q , then it holdsover all subfields of F n for all n . Incidentally, for p = 3 the “first layer”assumption is always satisfied (see Example 4.8).Using the above theorems, it is also possible to get a lower estimate onthe growth of the p ∞ -Selmer group in this tower by computing root num-bers. For example, if p ≡ E is semistable and admits arational p -isogeny, then combining Theorem 1.3 and [31] Cor. 2 shows thatdim X p ( E/F n ) ≥ ap n for some a > n .Finally, let us point out some of the things that definitely can not beobtained just from Theorem 1.6. It is tempting to try and prove the p -parityconjecture for A/K itself by finding a clever extension
F/K and a Gal(
F/K )-relation Θ with ∈ T Θ ,p . However, Theorem 2.56 shows that all τ ∈ T Θ ,p are even-dimensional (and have trivial determinant). So, even assumingfiniteness of X and using several primes p , one requires at least one addi-tional twist for which parity is known. For instance, the p -parity conjecturefor all elliptic curves over Q can be proved for odd p by reversing the argu-ment in 1.9 and 1.10: it is possible to find a suitable anticyclotomic extension EGULATOR CONSTANTS AND THE PARITY CONJECTURE 7 where one knows p -parity for the twists by ǫ and some 2-dimensional irre-ducible τ , whence it is also true for . (This is the argument used in [6].)It is also worth mentioning that if ρ is an irreducible Q p G -representationwhich is either symplectic or of the form σ ⊕ σ ∗ over ¯ Q p , then ( − h τ,ρ i = 1for every Θ and τ ∈ T Θ ,p , so Theorem 1.6 yields no information about theparity of such ρ in X p ( A/F ). Also, the theorem gives no interesting p -paritystatements when p ∤ | G | or G has odd order.For a summary of properties of τ ∈ T Θ ,p and examples see § Regulator constants and parity of Selmer ranks.
To explain ourapproach to the parity conjecture, let us first review the method of [6, 7]which allows one to express the Selmer parity in Theorem 1.6 in terms oflocal invariants of the abelian variety.Suppose
F/K is a Galois extension of number fields. For simplicity, con-sider an elliptic curve
E/K , and assume for the moment that the Tate-Shafarevich group X is finite. Define the Birch–Swinnerton-Dyer quotient
BSD(
E/K ) = Reg
E/K | X ( E/K ) | p | ∆ K | | E ( K ) tors | · C E/K , the conjectural leading term of L ( E/K, s ) at s = 1, see [40] §
1. Here Reg isthe regulator, C E/K = Q v C v ( E/K v , ω ) the product of local Tamagawanumbers and periods, and ∆ K is the discriminant of K (see § E i /K i are elliptic curves (or abelian varieties) that happen tosatisfy Q i L ( E i /K i , s ) n i = 1, then Q i BSD( E i /K i ) n i = 1 as predicted bythe Birch–Swinnerton-Dyer conjecture . Taking the latter modulo rationalsquares (to eliminate X and torsion) yields a relation between the regulatorsand the local terms C . It turns out, and has already been exploited in [5, 6],that this has strong implications for parities of ranks.As a first example, if E admits a K -rational p -isogeny E → E ′ , then theequality L ( E/K, s ) = L ( E ′ /K, s ) leads to the congruence C E/K C E ′ /K ≡ Reg E ′ /K Reg
E/K ≡ p rk( E/K ) mod Q × , where the second step is an elementary computation with height pairings.As a second example, if E/K is arbitrary and E d /K is its quadratic twist by d ∈ K × , then L ( E/K, s ) L ( E d /K, s ) = L ( E/K ( √ d ) , s ), and (cid:12)(cid:12)(cid:12)(cid:12) ∆ / K ( √ d ) ∆ K (cid:12)(cid:12)(cid:12)(cid:12) C E/K C E d /K C E/K ( √ d ) ≡ Reg
E/K ( √ d ) Reg
E/K
Reg E d /K ≡ rk( E/K ( √ d )) mod Q × . If Q L ( E i /F i , s ) = Q L ( E ′ j /F ′ j , s ), the corresponding products of Weil restrictionsto Q have the same L -function, hence isomorphic l -adic representations (Serre [35] § X is assumedfinite and BSD-quotients are invariant under Weil restriction (Milne [22] §
1) and isogeny(Tate–Milne [23] Thm. 7.3).
TIM AND VLADIMIR DOKCHITSER
The main subject of this paper is another massive source of identitiesbetween L -functions, relations between permutation representations. If F/K is a Galois extension with Galois group G , then a G -relationΘ : M i C [ G/H i ] ⊕ n i = 0 ( H i < G, n i ∈ Z )forces the identity Q L ( E/F H i , s ) n i = 1 by Artin formalism, which leads to Q ( C E/F Hi ) − n i ≡ Q (Reg E/F Hi ) n i mod Q × . By definition of the regulator,Reg
E/F Hi = det( | H i | h , i| ρ H i ) (= D ρ ( H i ) of § ρ = E ( F ) ⊗ Q and h , i is the height pairing on E/F . So the multiplic-ities rk σ ( E/F ) with which various irreducible Q G -representations σ occurin E ( F ) ⊗ Q satisfy Y i ( C E/F Hi ) − n i ≡ Y i (Reg E/F Hi ) n i ≡ C Θ ( ρ ) ≡ Y σ C Θ ( σ ) rk σ ( E/F ) mod Q × . In other words, the p -parts of the left-hand side determine the parities ofspecific ranks: for any τ p ∈ T Θ ,p , Y i ( C E/F Hi ) n i ≡ Y p p h τ p ,E ( F ) i mod Q × . The three procedures may be carried out without assuming that X isfinite, at the expense of working with Selmer groups rather than Mordell-Weil groups. In the first two cases, the outcome isdim X p ( E/K ) ≡ ord p C E/K C E ′ /K mod 2 (isogeny) , dim X ( E/K ( √ d )) ≡ ord C E/K C Ed/K C E/K ( √ d ) (cid:12)(cid:12)(cid:12) ∆ / K ( √ d ) ∆ K (cid:12)(cid:12)(cid:12) mod 2 (quad. twist) . In the case of G -relations, according to [7] Thms. 1.1, 1.5, we have Theorem 1.14.
Let
F/K be a Galois extension of number fields with Galoisgroup G . Let p be a prime and Θ = P n i H i a G -relation. For every ellipticcurve E/K , the Q p G -representation X p ( E/F ) is self-dual, and h τ, X p ( E/F ) i ≡ ord p Y i ( C E/F Hi ) n i mod 2 for all τ ∈ T Θ ,p . The same is true for principally polarised abelian varieties
A/K , except thatwhen p = 2 we require that the polarisation comes from a K -rational divisor. Remark 1.15.
In contrast to Theorem 1.6, this result has no constraints onthe reduction types of the abelian variety. So it always gives an expressionfor h τ, X p ( A/F ) i for τ ∈ T Θ ,p in terms of local data. Example 1.16.
As in Example 1.9, suppose Gal(
F/K ) = D p n . Then fora faithful 2-dimensional representation τ , the parity of h ⊕ ǫ ⊕ τ, X p ( A/F ) i is determined by local Tamagawa numbers, as ord p C A/F /C A/F C p mod 2.(Mazur and Rubin have another local expression for precisely the sameparity; see [20] Thm. A.) EGULATOR CONSTANTS AND THE PARITY CONJECTURE 9
Root numbers and Tamagawa numbers.
We have explained howin three situations ( p -isogeny, quadratic twist, G -relations) the parity ofsome Selmer rank can be expressed in terms of local Tamagawa numbers.As root numbers are also products of local root numbers, this suggests aproof of the corresponding case of the parity conjecture by a place-by-placecomparison (cf. [3, 5] for the isogeny case and [18, 19] for quadratic twists).There are two subtle points:First, the local terms do not always agree. In each case, one needs a goodexpression for the root numbers, separating the part that does agree with C v and an “error term” that provably dies after taking the product over allplaces. This error term in the isogeny case is an Artin symbol ([3] Thm. 2.7,[5] Thms. 3, 4), for quadratic twists it is a Legendre symbol ([19] p. 307),and in our case it comes out as the local root number w ( τ ) A (see The-orem 3.2). In fact, for group-theoretic reasons this contribution is trivial(Lemma A.1 and Theorem 2.56(1)), so here the local terms do agree.Second, although the remaining compatibility of the corrected local rootnumber and C v is a genuinely local problem, they are computed for com-pletely different objects — for instance in Example 1.16 the representation ⊕ ǫ ⊕ τ bears little resemblance to C E/F /C E/F C p . In the isogeny andquadratic twist cases, the proof of this compatibility in [3, 5, 18] boils downto brutally working out an explicit formula for each term separately. Thatthe two formulae then agree comes out as a miracle. In our case, for a fixed Galois group G and relation Θ this strategy works equally brutally,cf. [6] Prop. 3.3 for G = (cid:0) ∗∗ (cid:1) ⊂ GL ( F p ).The general case occupies § G -relations andregulator constants from §
2. We first reduce our “semilocal” problem (placescan split in
F/K ) to one about abelian varieties over local fields. If now
A/K is an abelian variety over a local field, in all cases covered by Theorem 1.6there is an explicit λ = ± K/K )-module V such that w ( A/K, τ ) = w ( τ ) A λ dim τ ( − h τ,V i for all self-dual τ (see Table 3.9). For instance, when A is semistable, λ = 1and V = X ( T ∗ ) ⊗ Q is the character group of the toric part of the reduction ofthe dual abelian variety (Proposition 3.26). The compatibility statement re-duces to proving that the function D V C v to Q × / Q × is representation-theoreticin the sense of § D X p / Q C v isrepresentation-theoretic. Thus V plays the rˆole of a “local version” of theSelmer module X p ( A/F ). Curiously, V is a rational representation, whichis only conjecturally true of the Selmer module. Example 1.17.
Take K = Q p for an odd prime p , and E/K an elliptic curvewith non-split multiplicative reduction of type I n . In this case the module V that computes root numbers is the 1-dimensional unramified character of order 2. Let us consider D V and C v (= c v , the local Tamagawa number) inthe unique C × C extension of Q p : nn (cid:0) ❅❅ (cid:0) C v ←−−−7 Q p ( √ u, √ p ) Q p ( √ u ) Q p ( √ up ) Q p ( √ p ) Q p ✟✟ ❍❍❍❍ ✟✟ D V d d (cid:0) ❅❅ (cid:0) Here Q p ( √ u ) is the quadratic unramified extension of Q p , and E has splitmultiplicative reduction precisely in those fields that contain it; d is thedeterminant of a fixed pairing on V , used in the definition of D V . Thegroup C × C has up to multiples just one relation (see Example 2.3),Θ = { } − C a − C b − C c + 2 C × C . The corresponding values of C v and D V are C v (Θ) = 2 n · (1 or 2) n · · · (cid:3) D V (Θ) = d · d · · , and so C v D V is representation theoretic (modulo squares!), by inspection.This example explains our need to understand G -relations, behaviour offunctions and D ρ . To establish the compatibility of local root numbers andTamagawa numbers in arbitrary extensions (Theorem 3.2), even for ellipticcurves with non-split multiplicative reduction, requires the full force of themachinery of § Notation.
For an abelian variety
A/K we use the following notation: X p ( A/F ) Hom Z p (lim −→ Sel p n ( A/F ) , Q p / Z p ) ⊗ Q p , the dual p ∞ -Selmer. w ( A/K ) local root number of
A/K for K local, orglobal root number, Q v w ( A/K v ) for K a number field. w ( A/K,τ ) (local/global) root number for the twist of A by τ , see [30]. c v ( A/K ) local Tamagawa number of A at a finite place v of K (when K is local, the subscript v is purely decorational). C v ( A/K, ω ) c v ( A/K ) ·| ω/ω o | K for K non-Archimedean, where | · | K isthe normalised absolute value, and ω o a N´eron differential; R A ( K ) | ω | for K = R ; 2 R A ( K ) ω ∧ ¯ ω for K = C .( K local; ω is a non-zero regular exterior form on A/K .) C A/K Q v C v ( A/K v , ω ) for any global non-zero regular exteriorform ω ; independent of ω (product formula).Notation for representations G → GL n ( K ), K a field of characteristic 0: h , i G , h , i usual inner product of two characters of representations; K [ G ]regular representation; K [ G/H ] permutation representation of G on the leftcosets of H ; trivial representation; τ ∗ contragredient representation; ρ H the H -invariants of ρ . We call ρ self-dual if ρ ∼ = ρ ∗ , equivalently ρ ⊗ ¯ K ∼ = ρ ∗ ⊗ ¯ K .For a K -vector space V and a non-degenerate K -bilinear pairing h , i withvalues in L ⊃ K , we write det( h , i| V ) ∈ L × / K × for det( h e i , e j i i,j ) in any K -basis { e i } of V . EGULATOR CONSTANTS AND THE PARITY CONJECTURE 11
For functions on subgroups of G we use the following notation:Θ a G -relation P i n i H i between permutation representations,i.e. P i C [ G/H i ] ⊕ n i = 0, see Definition 2.1. C Θ ( ρ ) regulator constant for a G -representation ρ , see Definition 2.13. ϕ (Θ) Q i ϕ ( H i ) n i for Θ = P i n i H i , see § ϕ ∼ ψ equivalence relation ϕ (Θ) = ψ (Θ) for all G -relations Θ, see § D ρ H det( | H | h , i| ρ H ), see Definition 2.40. · · · · see Definitions 2.33, 2.35.D n denotes the dihedral group of order 2 n (including C × C for n = 2).Conjugation of subgroups is usually written as a superscript, H x = xHx − .By a local field we mean a finite extension of Q p , R or F p (( t )) (the latterwill never occur). We write e M/L , f
M/L for the ramification degree and theresidue degree of an extension
M/L of local fields, µ n for the set of n th rootsof unity and ord p for the p -valuation of a rational or a p -adic number.2. Functions on the Burnside ring
This section is dedicated to relations between permutation representa-tions, behaviour of functions on the Burnside ring with respect to such rela-tions, the issue whether a function is representation-theoretic, and regulatorconstants. As explained in the introduction, the applications we have inmind relate to elliptic curves and abelian varieties. On the other hand, theresults are self-contained, purely group-theoretic in nature, and they maybe of independent interest.Throughout the section G is an abstract finite group. .2.i. Relations between permutation representations.
Let G be a finite group and S the set of subgroups of G up to conjugacy. Byabuse of notation, for a subgroup H < G we also write H for its class in S .The Burnside ring of G is the free abelian group Z S (we will not use its mul-tiplicative structure). The elements of S are in one-to-one correspondencewith transitive G -sets via H G/H . This extends to a correspondencebetween elements of Z S with non-negative coefficients and finite G -sets,under which addition translates to disjoint union.The map H C [ G/H ]( ∼ = Ind GH H ) defines a ring homomorphism fromthe Burnside ring to the representation ring of G . On the level of G -sets,this map is simply X C [ X ]. Here in § Definition 2.1.
We call an element of the Burnside ring of G Θ = X i n i H i ( n i ∈ Z , H i < G )a relation between permutation representations of G or simply a G -relation if ⊕ i C [ G/H i ] ⊕ n i = 0 as a virtual representation, i.e. the character P i n i χ C [ G/Hi ] is zero. In other words, if Θ corresponds to a formal difference of two G -sets,we require that they have isomorphic permutation representations. Exercise 2.2.
A cyclic group C n has no non-trivial relations. Example 2.3.
The group G = C × C has five subgroups { } , C a , C b , C c , G and four irreducible representations , ǫ a , ǫ b , ǫ c . Writing out the permutationrepresentations, C [ G ] ∼ = ⊕ ǫ a ⊕ ǫ b ⊕ ǫ c , C [ G/ C a ] ∼ = ⊕ ǫ a , C [ G/ C b ] ∼ = ⊕ ǫ b , C [ G/ C c ] ∼ = ⊕ ǫ c , ∼ = , we see that, up to multiples, there is a unique G -relationΘ = { } − C a − C b − C c + 2 G. Example 2.4.
Generally, any dihedral group G = D n with presentation h g, h | h n = g = ( gh ) = 1 i has the relation { } − h g i − h gh i − h h i + 2 G. When n is odd it can be written as { } − − C n + 2 D n , and it is uniqueup to multiples when n is prime. For D and D , together withD (cid:26) h g i−h gh i−h g, h i + h gh, h i{ }−h h i− h g i +2 h gh, h i D h h i−h h i− h g, h i +2 G h h i−h h i−h gh, h i−h g, h i +2 G h g i−h gh i + h gh, h i−h g, h i it forms a Z -basis of all G -relations (cf. Table 3.14 for the lattice of thesesubgroups.) Remark 2.5.
The number of irreducible Q G -representations coincides withthe number of cyclic subgroups of G up to conjugacy ([36] § G is not cyclic, this is clearly less than the number of all subgroups up toconjugacy, so the map P n i H i
7→ ⊕ C [ G/H i ] ⊕ n i must have a kernel. Henceevery non-cyclic group has non-trivial relations. Example 2.6 (Artin formalism) . Let
F/K be a Galois extension of numberfields with Galois group G . For H < G , the Dedekind ζ -function ζ F H ( s )agrees with the L -function over K of the Artin representation C [ G/H ]. Soa G -relation P i n i H i yields the identity Y i ζ F Hi ( s ) n i = 1 . Similarly, if
E/K is an elliptic curve (or an abelian variety), Y i L ( E/F H i , s ) n i = 1 . Notation 2.7.
For D < G , define a map Res D from the Burnside ring of G to that of D , and a map Ind GD in the opposite direction byRes D H = X x ∈ H \ G/D D ∩ H x − and Ind GD H = H .
On the level of representations (i.e. under H C [ G/H ]), these are the usualrestriction and induction. On the level of G -sets, Res D simply restricts theaction from G to D (Mackey’s decomposition). EGULATOR CONSTANTS AND THE PARITY CONJECTURE 13
Theorem 2.8.
Suppose
D, H i < G , N ⊳ G . (1) The sum and the difference of two G -relations is a G -relation. (2) If Θ = P i n i H i and m Θ is a G -relation, then Θ is a G -relation. (3) (lifting) If H i ⊃ N and P i n i H i N/N is a
G/N -relation, then P i n i H i is a G -relation. (4) (induction) Any D -relation is also a G -relation; i.e. if Θ = P i n i H i is a D -relation, then Ind GD Θ = P i n i H i is a G -relation. (5) (projection) If P i n i H i is a G -relation, then P i n i H i N/N is a
G/N -relation. (6) (restriction) If
Θ = P i n i H i is a G -relation, then its restriction Res D Θ = P i n i P x ∈ Hi \ G/D D ∩ H x − i is a D -relation.Proof. (1),(2),(3) Clear. (4) Induction is transitive. (5) This follows fromthe fact that C [ G/H i ] N ∼ = C [ G/N H i ] as a G -representation. (The invari-ants C [ G/H ] N come from orbits of N on G/H , so this space has a basis { P x ∈ ∆ xH } ∆ with ∆ ranging over the double cosets N \ G/H (= G/N H ).As N is normal, G permutes the basis elements, and this is the same asthe action on G/N H .) (6) This is a consequence of Mackey’s formula,Res D C [ G/H ] ∼ = ⊕ x ∈ H \ G/D C [ D/D ∩ H x − ]. (cid:3) Properties (3) and (4) allow one to lift relations from quotient groupsand induce them from subgroups. This is not to suggest that relations canalways be built up like that, for instance dihedral groups have relations whilecyclic groups do not. Here is a case when this does work:
Lemma 2.9.
Let
D ⊳ G , and suppose that G acts on the Burnside ring of D by conjugation through a quotient of order n . (1) If Θ = P i n i H i is a G -relation with H i ⊂ D , then n Θ is inducedfrom a D -relation. (2) Suppose that
N ⊳ G with N ⊂ D , and that each subgroup of G eithercontains N or is contained in D . Then for every G -relation Θ , n Θ is a sum of a relation induced from D and one lifted from G/N .Proof. (1) Let G be the kernel of the action of G on the Burnside ring of D .As a G -relation, we may write n Θ as n Θ = X i n i (cid:16) X g ∈ G/G gH i g − (cid:17) . We claim that in this form it is a D -relation. Indeed, restricting it to D ,on the one hand, yields a D -relation (Theorem 2.8(6)) and, on the otherhand, multiplies the expression by [ G : D ]. Hence the expression itself is a D -relation.(2) If Θ is a G -relation, write it as P i n i H i + P j n ′ j H ′ j with H i ⊃ N and H ′ j ⊂ D . ThenΘ = ( X i n i H i + X j n ′ j N H ′ j ) + ( X j n ′ j H ′ j − X j n ′ j N H ′ j ) . The first term is a relation lifted from
G/N (Theorem 2.8(5)), and the secondterm is therefore a G -relation with constituents in D , so (1) applies. (cid:3) Example 2.10.
Suppose G = C u m × C k with u odd and k > m . Set G = { }× C k − m ⊂ G = C u m × C k − ⊂ G, G/G ∼ = C u m × C m . Every element outside G generates a subgroup containing G , so everysubgroup not in G contains G . Since every subgroup of G is normal,Lemma 2.9(2) shows that every G -relation is a sum of a relation comingfrom G and one lifted from G/G . By induction, the lattice of G -relationsis generated by ones coming from subquotients C u m × C t / { }× C t − m for m ≤ t ≤ k , all isomorphic to C u × C m × C m . Example 2.11.
Suppose G = h x, y | x n = y k = 1 , yxy − = x − i , a semi-direct product of C k by C n for some k, n ≥
1. Consider G = h y i ⊂ G = h x, y i ⊂ G, G/G ∼ = D n , G/G ∼ = C . Note that if x a y b ∈ H < G with b odd, then ( x a y b ) = y b ∈ H implies that H ⊃ G . Equivalently, every subgroup not contained in G contains G . ByLemma 2.9(2), if Θ is any G -relation, then 2Θ is a sum of a relation inducedfrom G and one lifted from G/G . If 4 ∤ n , it is easy to verify that everysubgroup of G is normal in G , so Θ itself is already of this form.Observe that G ∼ = C n × C k − , whose relations were discussed in theprevious example.2.ii. Regulator constants.
Let K be a field of characteristic 0. In thissection we define regulator constants for self-dual K G -representations, firstintroduced in [6] for K = Q . (The name “regulator constant” comes fromregulators of elliptic curves; see § Notation 2.12.
Suppose V is a K -vector space with a non-degenerate K -bilinear pairing h , i that takes values in some extension L of K . We writedet( h , i| V ) ∈ L × / K × for det( h e i , e j i i,j ) in any K -basis { e i } of V . Definition 2.13.
Let G be a finite group, ρ a self-dual K G -representation,and Θ = P i n i H i a G -relation. Pick a G -invariant non-degenerate K -bilinearpairing h , i on ρ with values in some extension L of K , and define the regulatorconstant C Θ ( ρ ) = C K Θ ( ρ ) = Y i det( | H i | h , i| ρ H i ) n i ∈ K × / K × . (This is well-defined, non-zero and independent of h , i by Lemma 2.15 andTheorem 2.17. It follows that it lies in K × / K × rather than L × / K × as thepairing can be chosen to be K -valued.) Exercise 2.14.
Let G = C × C and Θ = { } − C a − C b − C c + 2 G fromExample 2.3. Then C Θ ( χ ) = 2 for all four 1-dimensional characters χ of G . EGULATOR CONSTANTS AND THE PARITY CONJECTURE 15
Lemma 2.15.
Suppose ρ is a K G -representation, and h , i a G -invariant K -bilinear non-degenerate pairing on ρ . For every H < G , the restriction of h , i to ρ H is non-degenerate. In other words, det( h , i| ρ H ) = 0 .Proof. Consider the projection P : ρ → ρ H given by v | H | P h ∈ H h · v .Then ρ = ρ H ⊕ ker P , and for v ∈ ρ H , w ∈ ker P , h v, w i = | H | X h ∈ H h hv, hw i = h v, P ( w ) i = 0 . So ρ H and ker P are orthogonal to each other, and the pairing cannot bedegenerate on either of them. (cid:3) Lemma 2.16.
Let
Θ = P i n i H i be a G -relation and ρ a K G -representation.Then X i n i dim ρ H i = 0 . Proof.
By Frobenius reciprocity, P n i dim ρ H i = P n i h Res H i ρ, H i i H i = P n i h ρ, Ind GH i H i i G = h ρ, ⊕ (Ind GH i H i ) ⊕ n i i G = h ρ, i G = 0 . (cid:3) We now prove that regulator constants are independent of the pairing:
Theorem 2.17.
Let
Θ = P i n i H i be a G -relation, ρ a self-dual K G -repre-sentation, and h , i , h , i two non-degenerate G -invariant K -bilinear pairingson ρ . Computing the determinants with respect to the same bases of ρ H i onboth sides, Y i det( | H i | h , i | ρ H i ) n i = Y i det( | H i | h , i | ρ H i ) n i . (This is an actual equality, not modulo K × .)Proof. We may assume K is algebraically closed. It is enough to prove thestatement for a particular choice of bases of ρ H i , as seen from the transforma-tion rule X M t XM for matrices of bilinear forms under change of basis.If ρ = α ⊕ β with α, β self-dual and Hom G ( α, β ∗ ) = 0, then h a, b i = 0 for a ∈ α and b ∈ β , and similarly for h , i . Since ρ H = α H ⊕ β H , choosing basesthat respect the decomposition reduces the problem to α and β separately.Thus, we may assume that either ρ = τ ⊕ n with τ irreducible and self-dual,or ρ = σ ⊕ n ⊕ ( σ ∗ ) ⊕ n with σ irreducible and not self-dual.In the first case, for each H i fix a basis of τ H i and take the induced basesof ( τ H i ) ⊕ n = ρ H i . Let h , i τ be a non-degenerate G -invariant bilinear pairingon τ , and let M i be its matrix on the chosen basis of τ H i . As h , i τ is uniqueup to scalar ( K is algebraically closed), the matrix of h , i on ρ H i is T (Λ , M i ) = λ M i λ M i . . . λ n M i λ M i λ M i . . . λ n M i ... ... . . . ... λ n M i λ n M i . . . λ nn M i , for some n × n matrix Λ = ( λ xy ) not depending on H i . Hencedet( | H i | h , i | ρ H i ) = (det Λ) dim τ Hi (det | H i | M i ) n . The dimensions dim τ H i cancel in Θ by Lemma 2.16, so Q i det( | H i | h , i | ρ H i ) n i does not depend on Λ, and takes therefore the same value for h , i .The argument in the second case is similar. The matrix of h , i on ρ H i isof the form (cid:0) T (Λ ′ ,M ′ i ) T (Λ ,M i )0 (cid:1) where M i and M ′ i are the matrices of a fixed G -invariant non-degenerate pairing h , i σ : σ × σ ∗ → K and its transpose.Again the contributions (( − n det Λ det Λ ′ ) dim τ Hi cancel in Θ and the resultfollows. (cid:3) Corollary 2.18.
Regulator constants are multiplicative in Θ and ρ , C Θ +Θ ( ρ ) = C Θ ( ρ ) C Θ ( ρ ) , C Θ ( ρ ⊕ ρ ) = C Θ ( ρ ) C Θ ( ρ ) . If L ⊃ K , then C K Θ ( ρ ) = C L Θ ( ρ ⊗ L ) in L × / L × . Example 2.19.
Suppose ρ = K [ G/D ] for some subgroup D of G . Take thestandard pairing on ρ , making the elements of G/D an orthonormal basis.The space of invariants ρ H has a basis consisting of H -orbit sums of thesebasis vectors. Since det( h , i| ρ H ) is the product of lengths of these orbits,det( | H | h , i| ρ H ) = Y w ∈ H \ G/D | H | | HwD || D | = Y w ∈ H \ G/D | H ∩ D w | , which yields an elementary formula for the regulator constants of ρ . Notethat for many groups, every K G -representation is a Z -linear combinations ofsuch ρ , e.g. dihedral groups D p n when K = Q or Q p , or symmetric groups. Example 2.20.
For an odd prime p , the dihedral group G = D p has therelation (cf. Example 2.4)Θ = { } − − C p + 2 G. For K = Q or Q p , the irreducible K G -representations are , sign ǫ and( p − ρ . Writing them as combinations of permutation repre-sentations K [ G/H ] = Ind GH H , we can compute their regulator constants asin Example 2.19: modulo squares, C Θ ( ) = C Θ (Ind GG ) = ( ) ( ) − ( p ) − ( p ) ≡ p C Θ ( ) C Θ ( ǫ ) = C Θ (Ind G C p ) = ( ) ( ) − ( p ) − ( p ) ≡ C Θ ( ) C Θ ( ρ ) = C Θ (Ind G C ) = ( p ) ( · ( p − / ) − ( ) − ( ) ≡ . So C Θ ( ) = C Θ ( ǫ ) = C Θ ( ρ ) = p . Example 2.21 (D p n , p odd, K = Q p ) . Generally, suppose G = D p n with p odd, and consider the G -relations coming from various D p subquotients,Θ n +1 − k = C p k − − p k − − C p k + 2 D p k (1 ≤ k ≤ n ) . EGULATOR CONSTANTS AND THE PARITY CONJECTURE 17
The irreducible Q p G -representations are , sign ǫ and ρ k of dimension p k − p k − for 1 ≤ k ≤ n . A computation as in Example 2.20 shows that C Θ k ( ) = C Θ k ( ǫ ) = C Θ k ( ρ k ) = p, C Θ k ( ρ j ) = 1 for j = k. Example 2.22 (D p n , p = 2, K = Q ) . For G = D n +1 consider the G -relationsΘ = C n − − D a n − D b n − C n + 2 G Θ n +1 − k = D a k − D b k − D a k +1 + D b k +1 (1 ≤ k < n ) . Here C k = h h n − k i , D a k = h h n − k , g i , D b k = h h n − k , gh i in terms of the gener-ators given in Example 2.4. In this case, the irreducible Q G -representationsare (trivial), ρ k of dimension 2 k − k − for 2 ≤ k ≤ n , and one-dimensionalcharacters ǫ, ǫ a , ǫ b that factor through G/ C n , G/ D a n − and G/ D b n − re-spectively. The regulator constants are C Θ ( ) = C Θ ( ǫ ) = C Θ ( ǫ a ) = C Θ ( ǫ b ) = 2 , C Θ k ( ǫ a ) = C Θ k ( ǫ b ) = C Θ k ( ρ k ) = 2 ( k > , and trivial on other irreducibles. Example 2.23.
Let G = SL ( F ), which is the semi-direct product of C by the quaternion group Q . Denote its complex irreducible representationsby , χ, ¯ χ (1-dim.), τ, χτ, ¯ χτ ( τ symplectic 2-dim.) and ρ (3-dim.). A basisof G -relations and their regulator constants for the Q G -irreducibles are χ ⊕ ¯ χ ρ τ ⊕ χτ ⊕ ¯ χτ C − C − Q + G − + 2Q Q is replaced by any K of characteristic 0, exceptthat τ may become realisable over K , in which case C K Θ ( τ ) and not just C K Θ ( τ ⊕ ) makes sense. This regulator constant will still be 1 by Corollary 2.25below, because τ is symplectic. Observe that in the table the representations V ⊕ V ∗ also have trivial regulator constants, which is true for all groups bythe same corollary.Let us record a number of situations when the regulator constants aretrivial; other properties are discussed in § Theorem 2.24.
Suppose ρ is a self-dual K G -representation such that ρ ⊗ K ¯ K admits a non-degenerate alternating G -invariant pairing. Then C Θ ( ρ ) = 1 for every G -relation Θ .Proof. Since dim Hom G ( , ρ ∧ ρ ) is the same over K and ¯ K , there is also a non-degenerate alternating G -invariant pairing h , i on ρ itself. By Lemma 2.15,its restriction to ρ H is non-degenerate (and alternating) for every subgroup H of G . In an appropriate basis for ρ H this pairing is given by a matrix (cid:0) − A t A (cid:1) , so det( | H | h , i| ρ H ) is a square in K . (cid:3) Corollary 2.25.
Let ρ be a self-dual K G -representation. Suppose either (1) ρ is symplectic as a ¯ K -representation, or (2) ρ ⊗ ¯ K ∼ = τ ⊕ τ for some ¯ K G -representation τ , or (3) all ¯ K -irreducible constituents of ρ ⊗ ¯ K are not self-dual.Then C Θ ( ρ ) = 1 for every G -relation Θ .Proof. It suffices to check that in each case ρ ⊗ K ¯ K carries a non-degeneratealternating G -invariant pairing. This holds by definition in case (1). Incases (2) and (3), ρ ⊗ K ¯ K is of the form V ⊕ V ∗ . Writing P for the matrixof the canonical map V × V ∗ → ¯ K , the pairing (cid:0) − P t P (cid:1) has the requiredproperties. (cid:3) Lemma 2.26.
Let ρ be a self-dual K G -representation. Suppose Θ = P n i H i is a G -relation such that no ¯ K -irreducible constituent of ρ occurs in any ofthe K [ G/H i ] . Then C Θ ( ρ ) = 1 .Proof. By Frobenius reciprocity, dim ρ H i = h , Res H i ρ i = h K [ G/H i ] , ρ i = 0. (cid:3) Remark 2.27.
Suppose R is a principal ideal domain whose field of fractions K has characteristic coprime to | G | . If ρ is a free R -module of finite rankwith a G -action, then C Θ ( ρ ) may be defined in the same way, C Θ ( ρ ) = Y i det( | H i | h , i| ρ H i ) n i ∈ K × /R × , where the determinants are now computed on R -bases of ρ H i . The pairing h , i may take values in any extension of K as before, and the class of C Θ ( ρ )in K × /R × is independent of h , i .For instance, when R = Z the group R × is trivial, so C Θ associates a well-defined rational number to every Z G -lattice. Also, if R is a discrete valuationring with maximal ideal m and residue field R/ m = k with char k ∤ | G | , it isnot difficult to see that C Θ ( ρ ⊗ K ) is in R × /R × , and C Θ ( ρ ⊗ K ) = C Θ ( ρ ⊗ k ) mod m . As every K G -representation admits a G -invariant R -lattice, we deduce Corollary 2.28.
Let K = Q or K = Q p , and let ρ be a K G -representation.If p ∤ | G | , then ord p C Θ ( ρ ) is even for every G -relation Θ . Functions modulo G -relations. We now turn to linear functions ϕ : Burnside ring of G −−−−→ abelian group (written multiplicatively) or, equivalently, functions on G -sets that satisfy ϕ ( X ∐ Y ) = ϕ ( X ) ϕ ( Y ). Ourmain concern is the distinction between functions that are representation-theoretic (i.e. only depend on C [ X ]) and those that are not. We say that • ϕ is trivial on an element Ψ of the Burnside ring of G if ϕ (Ψ) = 1. • ϕ ∼ ϕ ′ if ϕ/ϕ ′ is trivial on all G -relations.So, ϕ is representation-theoretic in the sense of § ϕ ∼ EGULATOR CONSTANTS AND THE PARITY CONJECTURE 19
Exercise 2.29.
For a constant λ , the function H λ [ G : H ] (= λ dim C [ G/H ] ) to R × is trivial on G -relations. On the other hand, H
7→ | H | in general is not. Example 2.30.
The constant function ϕ : H λ is trivial on G -relations: ϕ (cid:0)X n i H i (cid:1) = Y λ n i = Y λ n i h , C [ G/H i ] i = λ h , L C [ G/H i ] ⊕ ni i = λ = 1 . Example 2.31.
A cyclic group has no relations, so ϕ ∼ ϕ . Example 2.32. If E/K is an elliptic curve and G = Gal( F/K ), then L : H L ( E/F H , s )is a function with values in the multiplicative group of meromorphic func-tions on Re s > . By Artin formalism, L ∼ § C : H C E/F H and Reg : H Reg
E/F H satisfy C · Reg ∼ R × / Q × . Definition 2.33. If D < G , we say a linear function on the Burnside ring of G is local (or D -local ) if its value on any G -set only depends on the D -setstructure. Since G/H = ` x ∈ H \ G/D D/ ( H x − ∩ D ) as a D -set, this is equivalentto the following: there is a linear function ϕ D on the Burnside ring of D such that ϕ ( H ) = ϕ D (Res D H ) (cid:0) = Y x ∈ H \ G/D ϕ D ( H x − ∩ D ) (cid:1) . In this case we write ϕ = D, ϕ D G , or simply ϕ = D, ϕ D . Example 2.34.
Such functions arise naturally in a number-theoretic set-ting. Suppose
F/K is a Galois extension of number fields and v a placeof K . Write G = Gal( F/K ) and D = Gal( F z /K v ) for the local Galois groupat v (more precisely, a fixed decomposition group at v , so D < G ). UnderGalois correspondence, ϕ D associates something to every extension of K v ,in which case ϕ = D, ϕ D simply means ϕ ( L ) = Y places w | v in L ϕ D ( L w ) . (The double cosets HxD correspond to the places w of L = F H above v , and H ∩ D x are their decomposition groups in Gal( F/L ) = H .) Typical localfunctions are those counting primes w above v in F H , or primes with a givenresidue field F q : H λ { w above v in F H } ( = D, λ ) H λ { w above v in F H with k ( F Hw ) ∼ = F q } (cid:0) = D, H (cid:8) λ, , k ( F Hw ) ∼ = F q else (cid:9) (cid:1) , where k ( · ) denotes residue field. Another example is the function H Y w | v c w ( A/F H ) , that for an abelian variety A/K computes the product of local Tamagawa numbers in F H above v .Let I ⊳ D be the inertia subgroup. If a place w of F H corresponds to adouble coset HxD , its decomposition and inertia groups in
F/F H are H ∩ D x and H ∩ I x , respectively. Its ramification and residue degree over v are e w = | I || H ∩ I x | and f w = [ D : I ][ H ∩ D x : H ∩ I x ] (the order of Frobenius in F/K dividedby that in
F/F H ). Many of the local functions that we will consider in § e and f , which motivates the following definition. Definition 2.35.
Suppose
I ⊳ D < G with D/I cyclic, and ψ ( e, f ) is a func-tion of two variables e, f ∈ N . Define D, I, ψ : H Y x ∈ H \ G/D ψ (cid:0) | I || H ∩ I x | , [ D : I ][ H ∩ D x : H ∩ I x ] (cid:1) , the product being taken over any set of representatives of the double cosets.This is a D -local function on the Burnside ring of G , to be precise D, I, ψ = D, U ψ (cid:0) | I || U ∩ I | , | D || UI | (cid:1) . Theorem 2.36.
Let
I ⊳ D < G with D/I cyclic. Then ( ℓ ) (Localisation) If ϕ = D, ϕ D , and ϕ D is trivial on D -relations, then ϕ is trivial on G -relations. (q) (Quotient) If N ⊳ G and ϕ ( H ) = ϕ G/N ( HN/N ) for some function ϕ G/N on the Burnside ring of
G/N which is trivial on
G/N -relations,then ϕ is trivial on G -relations. (t) (Transitivity) If D < D < G , ϕ = D , ϕ G and ϕ = D , ϕ D ,then ϕ = D , ϕ G . (f) (Functions of f ) If ψ ( e, f ) does not depend on e , then D, I, ψ ∼ . (r) (Renaming) If I < I is normal in D with cyclic quotient, and ψ ( e, f ) is a function of the product ef , then D, I, ψ = D, I , ψ . (d) (Descent) If I < D < D and ψ ( e, f ) = ψ ( e, f /m ) m whenever m di-vides f and [ D : D ] , then D, I, ψ = D , I, ψ .Proof. ( ℓ ) and (q) follow from Theorem 2.8(6) and (5), respectively. (t) isimmediate from the G -set interpretation in Definition 2.33. (f) follows fromproperties ( ℓ ) and (q), and that the cyclic group D/I has no non-trivialrelations. (r), (d) follow from the definitions. (cid:3)
Example 2.37.
Property (f) shows that the two functions H λ ··· count-ing primes in Example 2.34 are representation-theoretic, in other words theycancel in relations. For instance, suppose that F/K is a Galois extension ofnumber fields and P i H i − P j H ′ j is a Gal( F/K )-relation. Writing L i = F H i and L ′ j = F H ′ j , X i { real places of L i } = X j { real places of L ′ j } . The same is true for complex places, or primes above a fixed prime v of K with a given residue degree over v . (This does not work when countingprimes with a given ramification degree instead.) EGULATOR CONSTANTS AND THE PARITY CONJECTURE 21
Example 2.38. If W < G is a subgroup of odd order, then W, W, e ∼ Q × / Q × . For W is solvable by the Feit-Thompson theorem,and picking W ⊳ W of prime index p , W, W, e = W, W , ef ∼ W, W , e = W , W , e . The assertion follows by induction. (The inductive step fails for p = 2, e.g.for G = C × C the function G, G, e does not cancel in the relation ofExample 2.3.)Finally, we record a variant of the “descent” criterion of Theorem 2.36d. Lemma 2.39 (Refined p -descent) . Let
N ⊳ G be of prime index p . Suppose φ, ψ are functions on the Burnside rings of G and N respectively. Then φ = N, ψ if and only if • φ ( H ) = ψ ( H ∩ N ) if H N , and • φ ( H ) = Q x ∈ G/N ψ ( xHx − ) if H ⊂ N , for any choice of representa-tives.Proof. The double cosets H \ G/N are precisely the left cosets
G/N for H ⊂ N , and there is a unique double coset otherwise. (cid:3) .2.iv. D ρ and T Θ ,p . We now introduce the function D ρ that computes regulator constants, anduse it to study their properties. Once again, K is any field of characteristiczero. At the end of the section, we reformulate these results for K = Q p interms of the sets T Θ ,p of § Definition 2.40.
For a self-dual K G -representation ρ with a non-degenerate K -valued G -invariant bilinear pairing h , i , define D ρ : H det( | H | h , i| ρ H ) ∈ K × / K × . By G -invariance of the pairing, D ρ ( H ) = D ρ ( xHx − ), so this is indeed afunction on the Burnside ring. If Θ is a G -relation, then by definition ofregulator constants D ρ (Θ) = C Θ ( ρ ) . Up to ∼ , this function is independent of the pairing: if D ′ ρ is defined in thesame way but with a different pairing on ρ , then D ρ ∼ D ′ ρ by Theorem 2.17.In particular, D ρ ⊕ ρ ′ ∼ D ρ D ρ ′ . Example 2.41. D is the function H
7→ | H | ∈ K × / K × , and D K [ G ] is theconstant function H Remark 2.42.
The function D ρ is representation-theoretic (i.e. ∼
1) ifand only if ρ has trivial regulator constants in all G -relations. For exam-ple, this happens if ρ carries a non-degenerate G -invariant alternating form(Theorem 2.24). On the other hand, C Θ ( ) need not be trivial, so D Lemma 2.43. If D < G and ρ is a self-dual K D -representation, then D Ind GD ρ ∼ D, D ρ as functions to K × / K × .Proof. Pick a D -invariant non-degenerate K -bilinear pairing h , i on ρ . For a D -set X , define a pairing ( , ) on Hom( X, ρ ) by( f , f ) = | D | X x ∈ X h f ( x ) , f ( x ) i , f , f ∈ Hom(
X, ρ ) . If X = D/U , the pairing ( , ) on Hom D ( X, ρ ) ⊂ Hom(
X, ρ ) agrees with | U | h , i on ρ U ⊂ ρ under the identification Hom D ( D/U, ρ ) = ρ U given by f f (1).So for a general D -set X = ` i D/U i , D ρ (cid:0)P i U i (cid:1) = det (cid:0) ( , ) (cid:12)(cid:12) Hom D ( X, ρ ) (cid:1) . Applying this to X = G/H , we have D, D ρ ( H ) = D ρ (Res D H ) = det (cid:0) ( , ) (cid:12)(cid:12) Hom D ( G/H, ρ ) (cid:1) = det (cid:0) | H | ( , ) (cid:12)(cid:12) [Hom D ( G, ρ )] H (cid:1) = D Ind GD ρ ( H ) , where the last equality uses that ( , ) is in fact a G -invariant pairing onHom D ( G, ρ ) = Ind GD ρ . (cid:3) Corollary 2.44.
Let
I ⊳ D < G with D/I cyclic. As functions to K × / K × , D K [ G/D ] 2.43 ∼ D, D ∼ D, H | H | ∼ D, H | D || H | = D, I, ef . Regulator constants behave as follows under lifting, induction and restric-tion of relations (cf. Theorem 2.8).
Proposition 2.45.
Let ρ be a self-dual K G -representation. (1) Suppose G = ˜ G/N and Θ is a G -relation. Lifting Θ to a ˜ G -relation ˜Θ ,we have C ˜Θ ( ρ ) = C Θ ( ρ ) . (2) If G < U and Θ is a U -relation, then C Θ (Ind UG ρ ) = C Res G Θ ( ρ ) . (3) If D < G and Θ is a D -relation, then C Θ (Res D ρ ) = C Ind GD Θ ( ρ ) .Proof. (1) The left- and the right-hand side are the same expression up to afactor of Q i | N | n i dim ρ Hi , if we write Θ = P i n i H i . This factor equals 1 byLemma 2.16. (2) This is a reformulation of Lemma 2.43. (3) Clear. (cid:3) In view of Example 2.41, the regular representation has trivial regulatorconstants. Generally, we have
Lemma 2.46. If H < G is cyclic, C Θ ( K [ G/H ]) = 1 for every G -relation Θ .Proof. For H cyclic, D K [ G/H ] 2.44 ∼ H, , f ∼ (cid:3) Theorem 2.47. If G has odd order, then C Θ ( ρ ) = 1 for every G -relation Θ and every self-dual K G -representation ρ . EGULATOR CONSTANTS AND THE PARITY CONJECTURE 23
Proof.
The only self-dual irreducible ¯ K G -representation is the trivial one.(Their number coincides with the number of self-inverse conjugacy classesof G , but these have odd order and so, except for the trivial class, have noself-inverse elements.) By Corollary 2.25(3), if ρ does not contain , then C Θ ( ρ ) = 1. But then C Θ ( ) = C Θ ( K [ G ]), which is 1 by Lemma 2.46. (cid:3) Corollary 2.48. C Θ ( K [ G/H ]) = 1 if H < G has odd order.Proof. C Θ ( K [ G/H ]) 2.45 (2) = C Res H Θ ( H ) = 1. (cid:3) Lemma 2.49.
Suppose K = Q or K = Q p , and N ⊳ H < G with H/N cyclic.If p ∤ | N | , then ord p C Θ ( K [ G/H ]) is even for every G -relation Θ .Proof. D K [ G/H ] 2.44 ∼ H, N, ef ∼ H, N, e and the values of e are divisorsof | N | . So C Θ ( K [ G/H ]) = D K [ G/H ] (Θ) has even p -valuation for p ∤ | N | . (cid:3) Reformulation for K = Q p . We now define the sets of representations T Θ ,p that encode those representations whose regulator constants are “ p -adiciallynon-trivial”. It is for these twists that we prove the p -parity conjecture(Theorem 1.6). So let us also restate the properties of regulator constantsin this language ( T Θ ,p -ese?) Definition 2.50.
Suppose K = Q p . For a G -relation Θ define T Θ ,p to bethe set of self-dual ¯ Q p G -representations τ that satisfy h τ, ρ i ≡ ord p C Θ ( ρ ) mod 2for every self-dual Q p G -representation ρ . Remark 2.51.
For instance, T Θ ,p contains representations of the form M ρ self-dual Q p -irr.ord p C Θ( ρ ) odd (any ¯ Q p -irreducible constituent of ρ ) . These are indeed self-dual, since C Θ ( σ ⊕ σ ∗ ) = 1 by Corollary 2.25. Notealso that these particular representations have no symplectic constituents orthose with even Schur index, by the same corollary.A general element of T Θ ,p differs from this one by something in T ,p , inother words by a self-dual (virtual) ¯ Q p -representation whose inner productwith any self-dual Q p -representation is even. Concretely, T ,p is generatedby representations of the form σ ⊕ σ ∗ (in particular σ ⊕ for self-dual σ ), andirreducible self-dual σ with either an even number of Gal( ¯ Q p / Q p )-conjugatesor even Schur index over Q p .In the context of the p -parity conjecture, the elements of T ,p correspondto twists τ for which the conjecture should “trivially” hold. The parity of h τ, X p ( A/F ) i is even, and we expect w ( A, τ ) = 1 for these τ . (This is indeedthe case whenever we have an explicit formula for this root number.) Example 2.52. If G = C n is cyclic, Θ = 0 is the only G -relation. The set T ,p consists of Z -linear combinations of ⊕ , sign ⊕ for n even, and χ ⊕ χ ∗ for all the remaining 1-dimensionals χ . Example 2.53. If G = D p with p odd, and Θ = { } − − C p + 2 G ofExample 2.20, then τ ∈ T Θ ,p if and only if τ contains an odd number oftrivial representations, an odd number of sign representations, and in totalan odd number of 2-dimensional ¯ Q p -irreducibles. Example 2.54. If G = D p n , and τ any 2-dimensional ¯ Q p G -representation,Examples 2.21, 2.22 show that τ ⊕ ⊕ det τ lies in T Θ ,p for some G -relation Θ. Example 2.55. If G = A , its complex- (or ¯ Q p -) irreducible representationsare , 3-dimensional τ , τ , 4-dimensional χ and 5-dimensional π . Here ⊕ π ∈ T Θ , ⊕ χ ⊕ π ∈ T Θ ′ , ⊕ τ i ⊕ χ ∈ T Θ ′′ , , for some G -relations Θ , Θ ′ , Θ ′′ (see [6] Ex. 2.19). Theorem 2.56 (Properties of T Θ ,p ) . Let Θ be a G -relation and τ ∈ T Θ ,p . (1) τ has even dimension and trivial determinant. (2) τ ⊕ τ ′ ∈ T Θ+Θ ′ ,p for τ ′ ∈ T Θ ′ ,p . (3) ˜ τ ∈ T ˜Θ ,p whenever G = ˜ G/N , and ˜ τ , ˜Θ are lifts of τ and Θ to ˜ G . (4) If D < G , then Res D τ ∈ T Res D Θ ,p . (5) If G < U , then Ind UG τ ∈ T Ind UG Θ ,p . (6) h τ, Q p [ G/H ] i is even whenever H < G is cyclic, has odd order orcontains a normal subgroup N ⊳ H with
H/N cyclic and p ∤ | N | . (7) If | G | is odd or coprime to p , then T Θ ,p = T ,p .Proof. (2) Clear.(3) Proposition 2.45(1) and Lemma 2.26.(4) Take any self-dual Q p D -representation ρ . Then modulo 2, h Res D τ, ρ i = h τ, Ind GD ρ i ≡ ord p C Θ (Ind GD ρ ) ≡ ord p C Res D Θ ( ρ ) . (5) Same computation, using Proposition 2.45(3).(6), (7) Reformulation of 2.28 and 2.46–2.49.(1) dim τ = h τ, Q p [ G ] i is even by (6). Now det τ ( g ) = 1 for all g ∈ G if andonly if det Res H τ = for all cyclic H < G . So (4) reduces the problem tocyclic groups, where it is clear (see Example 2.52). (cid:3) Corollary 2.57. 1 T Θ ,p and ⊕ ǫ T Θ ,p for any 1-dimensional ǫ = . Remark 2.58.
In view of Theorems 1.14 and 1.6 we may call T p = S Θ T Θ ,p the space of “ p -computable” twists. This set of representations is canonicallyassociated to a finite group G and a prime number p . It behaves well underrestriction and induction, and is closed under direct sums and tensor productwith permutation representations (since τ ⊗ Ind GH = Ind GH (Res H τ ) liesin T Ind GH Res H Θ ,p ). It would be very nice to have an intrinsic description of T p . EGULATOR CONSTANTS AND THE PARITY CONJECTURE 25 Root numbers and Tamagawa numbers
The aim of this section is to establish the following statement about com-patibility of local root numbers and local Tamagawa numbers. The proofwill occupy all of §§ p -parity conjecture. In fact, we expect the theorem below to hold forall principally polarised abelian varieties, and this would imply that therestrictions on the reduction of A in Theorem 1.6 could be removed. Notation 3.1.
Let K be a local field of characteristic zero, F/K a Galoisextension, and
A/K an abelian variety. For H < Gal(
F/K ) write (cf. § C v ( H ) = C v ( A/F H ) = C v ( A/F H , ω o )for any exterior form ω o on A/K , minimal if K is non-Archimedean. (Weinsist on minimality only for convenience: Theorem 3.2 below holds for anychoice of ω because C v ( · , ω ) ∼ C v ( · , ω o ), cf. proof of Corollary 3.4.) Theorem 3.2 (Existence of V ) . Let K be a local field of characteristiczero, F/K a Galois extension with Galois group D and A/K a principallypolarised abelian variety. Assume that either (1) D is cyclic, (2) A = E is an elliptic curve with semistable reduction, (3) A = E is an elliptic curve with additive reduction and K has residuecharacteristic l > , or (4) A/K has semistable reduction.Then there is a Q D -module V such that (Root) w ( A,τ ) w ( τ ) A = ( − h τ, V i for all self-dual representations τ of D , and (Tam) C v ∼ D V as functions on the Burnside ring of D . Equivalently, forevery D -relation Θ = P i n i H i , Y i C v ( A/F H i ) n i ≡ C Θ ( V ) mod Q × . In the following exceptional subcase of (4) we only claim (Tam) up to mul-tiples of 2: (4ex)
A/K is semistable, K has residue characteristic 2, the wild iner-tia group of F/K is non-cyclic and
A/K does not acquire splitsemistable reduction over any odd degree extension.
In the setting of the theorem, let Θ be a D -relation and p a prime number,odd in case (4ex). For any τ ∈ T Θ ,p , we obtain a chain of equalities: w ( A/K,τ ) w ( τ ) A (Root) = ( − h τ, V i = ( − ord p C Θ ( V⊗ Q p ) (Tam) = ( − ord p C v (Θ) (note that C v (Θ) ∈ Q × / Q × even for K = R , C by property (Tam), soord p C v (Θ) makes sense). By the determinant formula w ( τ ) = 1, as τ isself-dual and has trivial determinant (Theorem 2.56(1), Lemma A.1). Thus, Corollary 3.3 (Local compatibility) . Suppose
F/K and
A/K are as inTheorem 3.2. Let Θ be a D -relation and p a prime number, odd in case (4ex) .Then for every τ ∈ T Θ ,p , w ( A/K, τ ) = ( − ord p C v (Θ) . Now let us deduce Theorem 1.6. Suppose
F/K is a Galois extension of number fields , A/K an abelian variety and v a place of K . Fix a non-zeroregular exterior form ω on A/K , and define functions on the Burnside ringof Gal(
F/K ) by C w | v : H Y w | v C w ( A/F H , ω ) C : H C A/F H (cid:0) = Y v C w | v ( H ) (cid:1) , the first product taken over the places of F H above v . Corollary 3.4.
Let
F/K be a Galois extension of number fields,
A/K anabelian variety, and fix a place z of F above a place v of K . Suppose A/K v , F z /K v satisfy the assumptions of Theorem 3.2, and let p be a prime number,odd in case (4ex) . Then for every Gal(
F/K ) -relation Θ and τ ∈ T Θ ,p , w ( A/K v , Res
Gal( F z /K v ) τ ) = ( − ord p C w | v (Θ) . If the assumptions hold at all places v of K , then w ( A/K, τ ) = ( − ord p C (Θ) . Proof.
Write D = Gal( F z /K v ) < Gal(
F/K ) for the decomposition groupof z , and I for its inertia subgroup. First note that C w | v (Θ) is independentof the choice of the exterior form ω : if ω = αω ′ , then Q w | v C w ( A/F H , ω ) Q w | v C w ( A/F H , ω ′ ) = Y w | v | α | F Hw = D, I, | α | fK v ( H ) , and this function is trivial on Θ by Theorem 2.36f. So we may assume that ω is minimal at v .Now Res D τ ∈ T Res D Θ ,p by Theorem 2.56(4), so w ( A/K v , Res D τ ) = ( − ord p C v (Res D Θ) (Corollary 3.3)= ( − ord p C w | v (Θ) (as C w | v = D, C v , cf. 2.33, 2.34) . For the last claim, take the product over all places. (cid:3)
Proof of Theorem 1.6.
The abelian variety
A/K satisfies the hypothesis ofCorollary 3.4 at all places of K . So for every τ ∈ T Θ ,p w ( A/K, τ ) = ( − ord p C (Θ) 1.14 = ( − h τ, X p ( A/F ) i . (cid:3) EGULATOR CONSTANTS AND THE PARITY CONJECTURE 27
Setup.
In the remainder of § A/K and
F/K be as in the theorem, in particular K is again local . We split cases(1)-(3) into subcases and define an extension L of K as follows: Notation 3.5. (1) D is cyclic.(1-) | D | is odd; L = K .(1+) | D | is even; L is the unique quadratic extension of K inside F .(2) A = E is an elliptic curve with semistable reduction.(2G) E has good reduction; L = K .(2S) E has split multiplicative reduction; L = K .(2NS) E has non-split multiplicative reduction; L/K is quadratic un-ramified.(3) A = E is an elliptic curve with additive reduction, K has residuecharacteristic l >
3. Write ∆ E and c for the standard invariants ofsome model of E/K and e = , ord ∆ E ) .(3C) E has potentially good reduction, µ e ⊂ K ; L = K ( e √ ∆ E ) , a cyclic extension of K .(3D) E has potentially good reduction, µ e K ; L = K ( µ e , e √ ∆ E ), a dihedral extension of K .(3M) E has potentially multiplicative reduction; L = K ( √− c ).(4) A/K has semistable reduction; L is the smallest unramified extensionof K where A acquires split semistable reduction.We remind the reader that (4) has a subcase (4ex), see Theorem 3.2. Notethat (1) includes Archimedean places, and in (2)-(4) L is a minimal Galoisextension of K where A acquires split semistable reduction (cf. Lemma 3.22).In view of Lemma 3.8 below, we may and will henceforth assume Hypothesis 3.6. F contains L . Notation 3.7.
Henceforth write D = Gal( F/K ) ,D ′ = Gal( F/L ) ⊳ D,I = Inertia subgroup of D,W = Wild inertia subgroup of I. We work extensively with functions from the Burnside ring of D to Q × / Q × .For brevity, · · · stands for · · · D in §§ Lemma 3.8.
Suppose V is a Q G -module and M/K a Galois extensioncontained in F . If V satisfies (Root) and the p -part of (Tam) of Theorem 3.2,then W = V Gal(
F/M ) satisfies the same conditions for the extension M/K .Proof.
The irreducible constituents of W are precisely those of V that factorthrough Gal( M/K ), so W clearly satisfies (Root). If Θ is a Gal( M/K )-relation and ˜Θ is its lift to G (as in Theorem 2.8(3)), then C ˜Θ ( V ⊖ W ) = 1by Lemma 2.26. So W satisfies (Tam). (cid:3) The choice of V is forced by formulae for the local root numbers w ( A/K, τ ).For a self-dual representation τ of Gal( F/K ), we claim that w ( A/K, τ ) = w ( τ ) A λ dim τ ( − h τ,V i with λ = ± V of D/D ′ given in Table 3.9. HereCase D/D ′ λ V (1-) 1 w ( A/K ) 0(1+) C w ( A/K ) χ ⊕ b .... .... ... .... .... ... .... ... .... .... ... .... .... ... .... .... ... .... . (2G) 1 1 0(2S) 1 1 (2NS) C η .... .... ... .... .... ... .... ... .... .... ... .... .... ... .... .... ... .... . (3C) C , C , C , C ǫ , D , D − ǫ ⊕ η ⊕ σ (3M) C w ( χ ) χ .... .... ... .... .... ... .... ... .... .... ... .... .... ... .... .... ... .... . (4) cyclic 1 X ( T ∗ ) ⊗ Q Table 3.9.
Root numbers , χ, η and σ are the trivial character, the non-trivial character of order 2,the unramified quadratic character and the unique faithful 2-dimensionalrepresentation of D/D ′ . The exponent b is defined by ( − b = w ( A/K, ⊕ χ ) w ( χ ) A , ǫ is as in [30] Thm. 2 (we will not need it explicitly) and X ( T ∗ ) is thecharacter group of the torus in the Raynaud parametrisation of the dualabelian variety A t , see § V = V ⊕ (cid:26) , λ = 1 , Q [ D ] , λ = − . By Frobenius reciprocity, h τ, Q [ D ] i = dim τ , so V satisfies property (Root)of Theorem 3.2. Moreover, D Q [ D ] ∼ D V ∼ D V .It remains to prove the following Proposition 3.10.
In each of the cases (1)–(4) and
V, λ as in Table 3.9,we have D V ∼ C v (up to multiples of 2 in case (4ex) ) and w ( A/K, τ ) = w ( τ ) A λ dim τ ( − h τ,V i for every self-dual representation τ of Gal(
F/K ) . The proof is a case-by-case analysis and will occupy § § Case (1): Cyclic decomposition group.Lemma 3.11.
Suppose D = Gal( F/K ) is cyclic. Then w ( A/K, τ ) = w ( τ ) A w ( A/K ) dim τ if ∤ [ F : K ] ,w ( A/K, τ ) = w ( τ ) A w ( A/K ) dim τ (cid:0) w ( A/K, ⊕ χ ) w ( χ ) A (cid:1) h τ,χ i if | [ F : K ] . Moreover, w ( A/K ) and w ( A/K, ⊕ χ ) w ( χ ) A are ± . EGULATOR CONSTANTS AND THE PARITY CONJECTURE 29
Proof.
For the last claim, local root numbers of abelian varieties are ±
1, seee.g. [34] § w ( A/K ) = ±
1, and the same holds for the quadratic twistof A by χ . By the determinant formula, w ( χ ) = ± w ( A/K, ρ ⊕ ρ ∗ ) = 1 for every representation ρ , so itsuffices to check the formulae for τ = and for τ = χ when [ F : K ] is even.But this is clear as w ( ) = 1. (cid:3) As D is cyclic and has therefore no relations, we trivially have C v ∼ ∼ D V .This proves Proposition 3.10 in Case (1).3.iii. Case (2): Semistable elliptic curves.
The root number formulafollows from [30] Thm. 2 and the determinant formula (Lemma A.1).We now prove that D V ∼ C v . Note that the differential ω remains min-imal in all extensions of K , so C v ( E/F H ) = c v ( E/F H ) for all H < D . ByTate’s algorithm ([38], IV.9), in terms of e = e F H /K and f = f F H /K theseTamagawa numbers are:Reduction of E/K
Good Split I n Non-split I n c v ( E/F H ) 1 ne (cid:26) , ∤ f, ∤ ne, , ∤ f, | ne,ne, | f. (Case 2G) E has good reduction. D V = C v = 1.3.iii.2. (Case 2S) E has split multiplicative reduction. If E/K has type I n , D V = D Q [ D/D ] 2.44 ∼ D, I, ef ∼ D, I, ne = C v . (Case 2NS) E has nonsplit multiplicative reduction. If E/K hastype I n , then D V = D η ∼ D Q [ D/D ′ ] / D Q [ D/D ] 2.44 ∼ D ′ , I, ef / D, I, ef = D, I, n ef, ∤ f ( ef/ , | f o / D, I, ef = D, I, n , ∤ fef, | f o ∼ D, I, n , ∤ fen, | f o = C v · D, I, n , ∤ f, | ne , else o . It remains to show that the last factor is ∼
1. If n is even, it is a functionof f and therefore ∼ n is odd. Then D, I, n , ∤ f, | e , else o = I, I, n , | e , ∤ e o = I, W, n , | ef , ∤ ef o . If v ∤
2, then W has odd order, so this is a function of f , hence ∼ v |
2, so W is a 2-group and [ I : W ] is odd. Then I, W, n , | ef , ∤ ef o = W, W, n , | e , ∤ e o ∼ W, W, n , | e , ∤ e o . By Theorem 2.36 ℓ , it suffices to prove that the function on subgroups of Wϕ W : H n , H = W , H = W o is trivial on W -relations. Let W/ Φ = ¯ W be the maximal exponent 2 quotient of W (so Φ ⊳ W isits Frattini subgroup.) Since proper subgroups of W cannot have full imagein ¯ W , we have ϕ W ( H ) = ϕ ¯ W ( ¯ H ). By Theorem 2.36q, it is enough to verifythat ϕ ¯ W is trivial on ¯ W -relations. But every ¯ W -relation has an even numberof terms with H = ¯ W (only these have C [ ¯ W /H ] of odd dimension), so thisis clear.3.iv.
Case (3): Elliptic curves with additive reduction.
We now cometo the truly painful case of additive reduction. Thus l = 2 , K a finite extension of Q l and E/K has additive reduction. We write q forthe size of the residue field of K and δ for the valuation of the minimaldiscriminant of E/K . The asserted root number formula again comes from[30] Thm. 2, and it remains to show D V ∼ C v .Decompose the functions D V = a · d, C v = c v · ω with a ( H ) = det( h , i| V H ) d ( H ) = | H | − dim V H c v ( H ) = c v ( E/F H ) ω ( H ) = (cid:12)(cid:12)(cid:12) ω oE/K ω oE/F H (cid:12)(cid:12)(cid:12) F H . These are well-defined on conjugacy classes of subgroups of D and takevalues in Q × / Q × . The pairing h , i on V may be chosen arbitrarily, and wepicked one that seemed natural to give explicit values of a in Tables 3.14, 3.17(write V as a sum of permutation modules and use Example 2.19).The function ω may be expressed in terms of e = e F H /K and f = f F H /K as follows. The minimal discriminant of E/K has valuation δe over F H , so ω ( H ) = q ⌊ ( δe − δ H ) / ⌋ f where δ H is the valuation of the minimal discriminantof E/F H (cf. [37] Table III.1.2). If E has potentially good reduction, then0 ≤ δ H <
12. If the reduction is potentially multiplicative of type I ∗ n (so δ = 6 + n ), it becomes I ∗ ne over F H if e is odd ( δ H = 6 + ne ), and I ne if e iseven ( δ H = ne ). It follows easily that ω = (cid:26) D, I, q ⌊ δe ⌋ f , if E has potentially good reduction D, I, q ⌊ e ⌋ f , if E has potentially multiplicative reduction.3.iv.1. Reduction to 2-power residue degree.
Lemma 3.12. If K ′′ /K ′ is a subextension of F/K , and is unramified ofodd degree, then C v ( E/K ′′ ) ≡ C v ( E/K ′ ) mod Q × .Proof. For c v this follows from Lemma 3.22. For ω this is clear. (cid:3) Lemma 3.13.
It suffices to prove Case (3) of Proposition 3.10 when f F/K is a power of 2.Proof.
Let
N ⊳ D correspond to the maximal odd degree unramified exten-sion of K in F . As C v is unchanged in odd degree unramified extensions(Lemma 3.12), a repeated application of Lemma 2.39 with φ = ψ = C v EGULATOR CONSTANTS AND THE PARITY CONJECTURE 31 shows that C v = N, C v . We claim that D V ∼ N, D Res N V . Then, byTheorem 2.36 ℓ , it suffices to show that D Res N V ∼ C v as functions on theBurnside ring of N , as asserted.By Lemma 2.43, N, D Res N V ∼ D Ind GN Res N V . ButInd GN Res N V ∼ = V ⊗ Ind GN N ∼ = V ⊕ J with J of the form J ⊕J ∗ over ¯ Q . By Corollary 2.25, D J (Θ) = C Θ ( J ) = 1 forany D -relation Θ. Thus D J ∼
1, whence D Ind GN Res N V ∼ D V , as required. (cid:3) (Cases 3C, 3D) E/K has potentially good reduction.
By Lemma 3.13,it suffices to prove the following
Claim.
Suppose f F/K is a power of 2. Then c v ∼ a and ω ∼ d , and hence C v ∼ D V .Let e = e L/K be the ramification degree of L over K . The extension iseither cyclic or dihedral (cf. Table 3.9), to be precise(Case 3C) Gal( L/K ) ∼ = C e e = 2 , , , , (Case 3D) Gal( L/K ) ∼ = D e e = 3 , , . As E/L has good reduction, δ e ≡ e is the smallest suchinteger. Moreover, q ≡ ( − t mod e with t = 0 in Case 3C and t = 1 inCase 3D (see e.g. [30] Thm. 2).As V is a Gal( L/K )-representation, both a ( H ) and the exponent dim V H in d ( H ) depend only on F H ∩ L . In Case 3C, V = 0 and a ( H ) = d ( H ) = 1. ForCase 3D, we summarise in Table 3.14 the subgroups H (up to conjugacy) ofGal( L/K ) =
D/D ′ = D e = h g, h | h e = g = ( gh ) = 1 i with the following data: in the top row is H and its generators; in the Table 3.14.
Dihedral quotient e = 6 e = 4 e = 3 D g,h ∗ II , h , a g,h ∗ IV , b gh,h ∗ IV , g,h ∗ I ∗ , ?C h , a g , b gh , h ∗ , (cid:3)
13 I , D g,h ∗ III , a g,h ∗ I ∗ , h , b gh,h ∗ I ∗ , a g , h ∗ , b gh ,
111 I , D g,h ∗ IV , ?C h , g ,
113 I , ❛❛❛❛❛ ❅ (cid:0)✧✧✧✧✧✧✭✭✭✭✭✭✭✭✭✭✭✭✭✭◗◗ ❳❳❳❳❳❳❳ PPPPPPPP ◗◗✑✑✑✑✑✓ ❅ ❛❛❛❛❛❛ ❜❜ ✑✑✧✧ ◗◗✑✑✑ ❜❜❜PP ✧✧✑✑ PP bottom left corner a ( H ), followed by ∗ when dim V H is odd; in the bottomright corner the Kodaira symbol and c v for E/L H . The functions a and d are elementary to compute, and c v come from Lemma 3.22; (cid:3) denotes asquare value, ? an undetermined value, V = C × C . The entries 1(3) and3(1) for the Tamagawa numbers in the table for e = 6 mean that there areactually two tables, one with 1 and 3 and one with 3 and 1. (Similarly for2(4) and 4(2) when e = 4.) There are also identical tables, but with II, III,IV replaced everywhere by II ∗ , III ∗ and IV ∗ , respectively. Remark 3.15.
Note from the pictures that c v ∼ a in Gal( L/K )-relationsin Case D; see Example 2.4 for the list of relations. This is also true inCase C, as Gal(
L/K ) is cyclic and has no relations. Now, a is a functionlifted from Gal( L/K ). If c v were such a function as well, we would have c v ∼ a in general by Theorem 2.36q. Unfortunately, life is not that simple,and we will use the full force of the machinery in § Proposition 3.16. c v ∼ a .Proof. We proceed as follows
Step 1: Reduction to D = C e ⋊ C k .We claim that a and c v are both lifted from Gal( L u /K ), where we use u todenote the maximal unramified extension in F . Then by Theorem 2.36q wemay replace F by L u , and we will be left with the case e F/K = e and D = C e ⋊ C k .That a lifts from Gal( L u /K ) is clear. In view of Lemma 3.22, to see that c v has this property it is enough to check that for every intermediate field M of F/K , the extension
M/M ∩ L u is totally ramified with gcd( e M/K , e ) =gcd( e M ∩ L u /K , e ). By Lemma 3.24, there is a subfield K ⊂ M e ⊂ M with M/M e totally ramified and e M e /K = gcd( e M/K , e ); so it suffices to show that L u ∩ M contains M e , equivalently that M e ⊂ L u . But M u e /K u and L u /K u sit inside the (cyclic) maximal tame extension F W /K u , so M u e ⊂ L u bycomparison of degrees.We now have D = C e ⋊ C k = h x, y | x e = y k = 1 , yxy − = x ± i with x +1 in Case 3C and x − in Case 3D, I = C e = h x i and e = 2 , , , Step 2: Proof for e = 2 , , e = 2 , ∤ e ), it is enough to prove that c v /a is trivial on relations of all the subquotients H t /N t where H t = h x, y t i , N t = h y t +1 i . In Case 3D, the t = 0 quotient is D/D ′ ∼ = D e , where c v /a does cancel in rela-tions by Remark 3.15. We consider the remaining subquotients ( ∼ = C e × C )in Cases C and D, according to the value of e . If e = 3, these are cyclic andhave no relations, so the result is trivial. If e = 6, H t /N t has the followinglattice of subgroups. Here we specify the group name, its generators ( h isthe image of x , and g a suitable element of order 2), the value of a ( X = 1 in EGULATOR CONSTANTS AND THE PARITY CONJECTURE 33
Case 3C, and X = 3 in Case 3D), the reduction type and Tamagawa numbersover the corresponding fields: C × C g,h , c gh , b gh , a h , h , g,h ∗ , ? C c gX I , b gh , a h ∗ , (cid:3) X I , ✟✟❜❜❜ ❜❜❜✄✄✧✧✧✑✑◗◗◗ ❜❜❜✧✧✧ PPPPPPPPPP PPPPPPPPPPP PPPPPPPPPP PPPPPPPPPPP ❳❳❳❳❳❳❳❳❳❳❳ Table 3.17. C × C - subquotientThe lattice of relations is generated by C − C a − C b − C c + 2(C × C ) and1 − C a − C b − C c + 2V, on each of which a/c v is trivial by inspection. Finallyif e = 2, the subquotients are isomorphic to C × C , and the data for theirsubgroups is the same as for the subgroups of V ⊂ C × C in Table 3.17with X = 1. Once again, a/c v is trivial on the unique C × C relation, whichcompletes the proof that c v ∼ a for e = 4. Step 3: Proof for e = 4.Suppose e = 4 and we are in Case 3D. By Example 2.11 and the proof ofLemma 2.9(1), every D -relation is a sum of one lifted from D/D ′ ∼ = D anda D -relation with terms in U = h x, y i . By Remark 3.15, c v /a cancels inrelations of D/D ′ , so it suffices to prove cancellation in D -relations whoseterms lie in U . Subgroups of U project to subgroups of C in D/D ′ , so a = 1for these (see Table 3.14). Hence it is enough to show that the followingfunction cancels in D -relations with terms in U :˜ c v ( H ) = c v ( E/F H ) · | H | , H ⊂ h y i , , h x i ⊂ H ⊂ h x , y i , , otherwise . As h x i , h y i and h x , y i are normal in D , the “correction terms” are thesame for conjugate subgroups, so this is a function on the Burnside ringof D . Exceptionally, we view ˜ c v as a function to R > , not (!) to Q × / Q × .The point is that now it suffices to check that it cancels in U -relations, sincea multiple of a D -relation with terms in U is an U -relation by Lemma 2.9(1),and taking roots is perfectly ok in R > .By Example 2.10, U -relations are generated by those coming from thesubquotients H t /N t (1 ≤ t ≤ k −
2) with H t = h x, y t i , N t = h y t +2 i , H t /N t ∼ = C × C . In such a subquotient, let g and h be the images of x and y t respectively.Note that the field F H t ( √ ∆ E ) corresponds to h g , h i . (By Lemma 3.22, itdistinguishes between c v = 2 and c v = 4 in the I ∗ -case; c v = 1 cannot occur because E/K has c v = 2 and so has non-trivial 2-torsion in every extensionof K , see Remark 3.23.) The subquotient H t /N t has a basis of 5 relations,given below with the corresponding ˜ c v : h g i−h gh i−h h , g i−h g i +2 h g, h i → –1 –1 –1 = 1 h h , g i−h g , h i−h g , gh i−h g, h i +2 h g, h i → –1 –1 –1 = 1 h h g i−h g h i−h h , g i−h gh i +2 h g , gh i → –1 –1 –1 = 1 { }−h g i−h h g i−h h i +2 h h , g i → (4 k - t -2 ) –1 –1 (4 k - t -1 ) –1 = 1 h h i−h h , g i−h h i−h g h i +2 h g , h i → (4 k - t -1 ) –1 (4 k - t ) –1 –1 = 1 . As the ˜ c v cancel in all relations, this proves the claim in Case 3D.Finally, in Case 3C the a and the c v are the same as for the subgroup H of Case 3D, so they again cancel in relations. (cid:3) Proposition 3.18.
In Case 3C, ω ∼ . In Case 3D, ω ∼ I, W, (cid:8) e, , e ∤ f e | f (cid:9) .Proof. As before, write δ for the valuation of the minimal discriminant of E/K and q for the size of the residue field of K . So q ≡ ( − t mod e with t = 0 in Case 3C and t = 1 in Case 3D. If q is an even power of the residuecharacteristic l , then t = 0 and ω = q ... = 1 ∈ Q × / Q × as asserted. Supposefrom now on that q is an odd power of l , so ω = D, I, q ⌊ δe ⌋ f = D, I, l ⌊ δe ⌋ f = I, I, l ⌊ δe ⌋ = I, W, l ⌊ δef ⌋ . The order of W is a power of l , so let us define k by e = l k .Define n ∈ { , } by lδ ≡ qδ ≡ ( − t δ + 12 n mod 24 , so l k δ ≡ ( − tk δ + 12 nk mod 24 . Then ω = I, W, l ⌊ δlkf ⌋ = I, W, l ⌊ ( − tkδf + fkn ⌋ = I, W, l ⌊ ( − tkδf ⌋ I, W, e fn . The second term is trivial: I, W, e fn = W, W, e n ∼ . As for the first term, for t = 0 it is a function of f so it is ∼ ⌊ m ⌋ ≡ ⌊ − m ⌋ mod 2 if and only if 12 | m ,so for t = 1 we have I, W, l ⌊ ( − kδf ⌋ = I, W, l ⌊ δf ⌋ I, W, n , | k , | δfl, else o ∼ I, W, n , | k , | δfl, else o = I, W, n l k , | k , e | fl k , else o = I, W, (cid:8) e, , e ∤ f e | f (cid:9) , as asserted. (cid:3) Proposition 3.19.
In Case 3C, d ∼ . In Case 3D, d ∼ I, W, (cid:8) e, , e ∤ f e | f (cid:9) .Proof. In Case 3C, the module V is zero and the result is trivial, so supposewe are in Case 3D. By definition, d ( H ) is either 1 or | H | (up to squares),depending on the intersection of F H with the dihedral extension L/K . We
EGULATOR CONSTANTS AND THE PARITY CONJECTURE 35 may replace | H | by [ D : H ] = e F H /K f F H /K by Lemma 2.16. Inspecting ∗ inTable 3.14, we find that for H ⊃ D ′ (i.e. corresponding to subfields of L ),( ef ) dim V H ≡ n , | f or e | eef, else mod Q × (with e = e F H /K , f = f F H /K ) . By Lemmas 3.24, 3.25, the condition “2 | f or e | e ” holds for a general F H ifand only if it holds for F H ∩ L . Therefore d ∼ ( H ( e F H /K f F H /K ) dim V H ) = D, I, n , | f , e | eef, else o . Recall that 2 , ∤ | W | and that [ D : I ] is a power of 2 (Lemma 3.13). Thus D, I, n , f =11 , e | ee, else o = I, I, n , e | ee, e ∤ e o = I, W, n , e | fef, e ∤ f o ∼ I, W, n , e | fe, e ∤ f o . (cid:3) Combining Propositions 3.16, 3.18 and 3.19 proves Proposition 3.10 inCases (3C) and (3D).3.iv.3. (Case 3M)
E/K has potentially multiplicative reduction.
In view ofLemma 3.13, it suffices to prove the following
Claim.
Suppose f F/K is a power of 2. Then ω ∼ ∼ a and c v ∼ d , andhence C v ∼ D V . Proposition 3.20. ω ∼ ∼ a .Proof. As V is a C -representation and C has no relations, a ∼ ω = D, I, q ⌊ e ⌋ f ∼
1, by the proof of e = 2 , δ = 6 caseof Proposition 3.18. (cid:3) Proposition 3.21. c v ∼ d .Proof. For H < D , the expression for c v ( E/F H ) is given in Lemma 3.22 (notethat L = K ( √− B ) in its notation). Writing e = e F H /K , f = f F H /K , wehave E/F H c v d c v /dL ⊂ F H I ne split ne | D | ef n | D | e fL F H , | e I ne non-split 2 1 2 L F H , ∤ e, f = 1 I ∗ ne c v ( E/K ) 1 c v ( E/K ) L F H , ∤ e, | f I ∗ ne L u and K u for the maximal unramified extensions of L and K inside F ,respectively. By Lemma 3.24, the function c v /d (to Q × / Q × ) factors throughGal( L u /K ) = Gal( L u /K u ) × Gal( L u /L ) = C × C k . By Theorem 2.36q, it suffices to prove that c v /d is trivial on Gal( L u /K )-relations. A list of generating relations is given in Example 2.10 (with u = 1and m = 1). These come from C × C -subquotients, each with one relation(Example 2.3) and it is elementary to verify that c v /d is trivial on these. (cid:3) Appendix: Local lemmas.
Lemma 3.22.
Let K ′ /K/ Q l be finite extensions with l ≥ , and let E/K bean elliptic curve with additive reduction, E : y = x + Ax + B, A, B ∈ K. Write ∆ = − A + 27 B ) for the discriminant of this model, ð for its K -valuation, and e = e K ′ /K .If E has potentially good reduction, then gcd( ð e,
12) = 2 = ⇒ c v ( E/K ′ ) = 1 (II , II ∗ )gcd( ð e,
12) = 3 = ⇒ c v ( E/K ′ ) = 2 (III , III ∗ )gcd( ð e,
12) = 4 = ⇒ c v ( E/K ′ ) = n , √ B K ′ , √ B ∈ K ′ (IV , IV ∗ )gcd( ð e,
12) = 6 = ⇒ c v ( E/K ′ ) = n , √ ∆ K ′ , √ ∆ ∈ K ′ (I ∗ )gcd( ð e,
12) = 12 = ⇒ c v ( E/K ′ ) = 1 (I ) . The extension K ′ ( √ B ) /K ′ in the IV , IV ∗ cases and K ′ ( √ ∆) /K ′ in the I ∗ case is unramified. In particular if K ′′ /K ′ has odd residue degree and gcd( ð e K ′ /K ,
12) = gcd( ð e K ′′ /K , , then c v ( E/K ′ ) = c v ( E/K ′′ ) .If E has potentially multiplicative reduction of type I ∗ n over K , then ∤ e, ∤ n = ⇒ c v ( E/K ′ ) = n , √ B K ′ , √ B ∈ K ′ (I ∗ ne )2 ∤ e, | n = ⇒ c v ( E/K ′ ) = n , √ ∆ K ′ , √ ∆ ∈ K ′ (I ∗ ne )2 | e, √− B K ′ = ⇒ c v ( E/K ′ ) = 2 (I ne non-split )2 | e, √− B ∈ K ′ = ⇒ c v ( E/K ′ ) = ne (I ne split ) . The extension K ′ ( √ B ) /K ′ , K ′ ( √ ∆) /K ′ or K ′ ( √− B ) /K ′ correspondingto the case is unramified.Proof. This follows from Tate’s algorithm ([38], IV.9). (cid:3)
Remark 3.23.
Let K be a finite extension of Q l with l ≥
5, and
E/K an elliptic curve. Recall that c v ( E/K ) is the size of the N´eron componentgroup E ( K ) /E ( K ). When E/K has additive reduction, this group is oforder at most 4, and is isomorphic to the prime-to- l torsion in E ( K ). (Usethe standard exact sequence [37] VII.2.1 and note that multiplication-by- l isan isomorphism both on the formal group of E and on the reduced curve.)In particular, in the I ∗ case of the lemma, c v ( E/K ′ ) = 1 if and only if E/K ′ has trivial 2-torsion. Lemma 3.24.
Suppose
L/K/ Q l are finite extensions. For every divisor m | e L/K with l ∤ m , there exists a subfield M of L/K with e M/K = m and L/M totally ramified.Proof.
Replacing K by its maximal unramified extension in L , we may as-sume that L/K is totally ramified. Let F be the Galois closure of L/K , andwrite G = Gal( F/K ), H = Gal( F/L ) and I for the inertia subgroup of G .Let N be the unique index m subgroup of I . We claim that M = F NH will do. EGULATOR CONSTANTS AND THE PARITY CONJECTURE 37
Since e M/K = [ I : I ∩ N H ] and [ I : N ] = m , it is enough to prove that N = I ∩ N H . “ ⊂ ” is clear. Observe that every subgroup U of I whoseindex is divisible by m is contained in N . (This is clear if U contains thewild inertia subgroup W ⊳ I , as
I/W is cyclic; otherwise replace U by U W .)In particular, m | e L/K = [ I : I ∩ H ] implies that I ∩ H ⊂ N . Since N ischaracteristic in I ⊳G and therefore normal in G , it follows that I ∩ N H ⊂ N as asserted. (cid:3) Lemma 3.25.
Suppose K/ Q l is finite, e = 2 , , , , and the size of theresidue field of K is congruent to − modulo e . Then the compositum F of all totally ramified extensions of K of degree e is a dihedral extension ofdegree e . Specifically, F = K ′ ( e √ π ) with π a uniformiser of K and K ′ /K quadratic unramified.Proof. For e = 2 this is elementary. Otherwise, K ( µ e ) = K ′ and it sufficesto prove that every totally ramified degree e extension of K is contained in K ′ ( e √ π ).For e = 3 suppose L/K is cubic totally ramified. It cannot be Galois bylocal class field theory, since the units of K have no index 3 subgroups. So itsGalois closure is an S -extension, which is tame and so contains K ′ = K ( µ ).By Kummer theory, LK ′ /K ′ is contained in the C × C -extension M of K ′ obtained by adjoining cube roots of all elements of K ′ . But it is easy to seethat Gal( M/K ) ∼ = C ⋉ C has a unique S -quotient, so LK ′ = K ′ ( e √ π ) and L ⊂ K ′ ( e √ π ) as asserted.For e = 4, every totally ramified quartic extension of K has a quadraticsubfield by Lemma 3.24, so there are at most 4 of them by the e = 2 case.Since in this case K ′ ( e √ π ) has 4 totally ramified subfields corresponding tothe non-normal subgroups of order 2 in D , they are all contained in it.For e = 6 the assertion follows from the e = 2 and e = 3 cases (applyLemma 3.24 for m = 2 and m = 3). (cid:3) Case (4): Semistable abelian varieties.
Review of abelian varieties with semistable reduction.
Let K be a fi-nite extension of Q l , let A/K be an abelian variety and take a prime p = l .Fix a finite Galois extension L/K where A acquires split semistable reduc-tion. By the work of Raynaud ([27], [13] § A / O L , which is an extension0 −→ T −→ A −→ B −→ , with T / O L a split torus and B / O L an abelian scheme, and such that A ⊗ ( O L /m iL ) is the identity component of the N´eron model of A over O L base changed to O L /m iL . These properties characterise A up to a uniqueisomorphism. In particular, the group Gal( L/K ) acts naturally on T , A and B . The character group of T is a finite free Z -module with an action ofGal( L/K ), and we denote it X ( T ). The dual abelian variety A t /K also has split semistable reduction over L ,and there is a sequence as above with T ∗ , A ∗ and B ∗ ∼ = B t ([13] Thm. 5.4).Raynaud constructs a canonical map X ( T ∗ ) ֒ → A ( L ), inducing an isomor-phism of Gal( ¯ K/K )-modules A ( ¯ K ) ∼ = A ( ¯ K ) /X ( T ∗ ) , which generalises Tate’s parametrisation for elliptic curves. (The Gal( ¯ K/K )-action on A ( ¯ K ) comes from the Galois action of Gal( ¯ K/L ) and the geomet-ric action of Gal(
L/K ); see [3] § p -adic Tate modules of the generic fibres over L ,0 −→ T p ( T L ) −→ T p ( A L ) −→ T p ( B L ) −→ −→ T p ( A L ) −→ T p ( A ) −→ X ( T ∗ ) ⊗ Z p −→ . In particular, T p ( A ) has a filtration with graded piecesgr = X ( T ∗ ) ⊗ Z p , gr = T p ( B L ) , gr = Hom( X ( T ) , Z p (1)) . Now suppose
A/K has semistable reduction. The reduction becomes splitsemistable over some finite unramified extension of K , and we take L to bethe smallest such field; so now Gal( L/K ) is cyclic, generated by Frobenius.To describe the Tamagawa number c v ( A/K ) and the action of inertia on T p ( A ) we use the monodromy pairing X ( T ∗ ) × X ( T ) −→ Z . This is a non-degenerate Gal(
L/K )-invariant pairing, and induces a Galois-equivariant inclusion of lattices N : X ( T ∗ ) ֒ −→ Hom( X ( T ) , Z ) . These have the same Z -rank, so N has finite cokernel. Moreover, N is co-variantly functorial with respect to isogenies of semistable abelian varieties.Any polarisation on A gives a map X ( T ∗ ) → X ( T ), and the induced pairing X ( T ∗ ) × X ( T ∗ ) −→ Z is symmetric ([13] § A is principally polarised, we geta perfect Galois-equivariant symmetric pairingcoker N × coker N −→ Q / Z . If K ′ /K is a finite extension, then X ( T ) and X ( T ∗ ) remain the same mod-ules (restricted to Gal( LK ′ /K ′ ) ⊂ Gal(
L/K )) by uniqueness of Raynaudparametrisation. The map N becomes e K ′ /K N , see [13] 10.3.5.The Gal( ¯ K/K )-module gr ⊕ gr ⊕ gr is unramified and semisimple, soit is a semisimplification of T p A . With respect to this filtration, the inertiagroup acts on T p A by I ¯ K/K ∋ σ t p ( σ ) N ∈ Aut T p ( A ) EGULATOR CONSTANTS AND THE PARITY CONJECTURE 39 with t p : I ¯ K/K → Z p (1) defined by σ σ ( π /p n K ) /π /p n K ∈ µ p n for anyuniformiser π K of K ([13] §§ A/ O K . It is an ´etale group scheme over the residuefield k of K , so Φ( k ) = Φ(¯ k ) Gal(¯ k/k ) consists of components defined over k .As K is complete and k is perfect, by [2] Lemma 2.1 the natural reductionmap A ( K ) → Φ( k ) is onto, so c v ( A/K ) = | Φ( k ) | . Finally, by [13] Thm. 11.5,Φ = coker N as groups with Gal(¯ k/k ) = Gal( K un /K )-action, so c v ( A/K ) = | (coker N ) Gal(
L/K ) | . Local root numbers for twists of semistable abelian varities.
Proposition 3.26.
Suppose
A/K is semistable, let
F/K be a finite Galoisextension containing L , and τ a complex representation of Gal(
F/K ) . Then w ( A/K, τ ) = w ( τ ) A ( − h τ,X ( T ∗ ) i . Proof.
Let H ( A ) = H ( A, Z p ) ⊗ Z p C = Hom( T p A ⊗ C , C ) for some p = l ,and let H ( A ) ss be its semisimplification. Write V = X ( T ∗ ) ⊗ Q , andsgn z = z/ | z | for z ∈ C × . By the unramified twist formula [39] 3.4.6, w (( H ( A ) ss ) ⊗ τ ) = w ( τ ) A sgn det(F | H ( A ) ss ) ν for some integer ν and F = Frob − K/K . Since det( H ( A )) is a power of thecyclotomic character, this expression is just w ( τ ) A . By [39] 4.2.4, w ( A/K, τ ) = w ( H ( A ) ⊗ τ )= w (( H ( A ) ss ) ⊗ τ ) sgn det( − F | (( H ( A ) ss ) ⊗ τ ) I )sgn det( − F | (( H ( A ) ⊗ τ ) I ) = w ( τ ) A sgn det( − F | gr ∗ ⊗ τ I )= w ( τ ) A sgn det( − F | V ⊗ τ I )= w ( τ ) A ( − dim V dim τ I det(F | V ) dim τ I det(F | τ I ) dim V , where the penultimate equality again comes from [39] 3.4.6. If η denotes theunramified character F
7→ −
1, we have( − dim V = ( − h ,V i + h η,V i , det(F | V ) = ( − h η,V i and similarly for τ I in place of V , as they are both self-dual and unramified.Now a trivial computation shows that( − h τ,V i = ( − h τ I ,V i = ( − h ,τ I ih ,V i + h η,τ I ih η,V i coincides with w ( A/K, τ ) /w ( τ ) A , as asserted. (cid:3) Tamagawa numbers for semistable abelian varieties.
Proposition 3.27.
Suppose
A/K is semistable and principally polarised,and set V = X ( T ∗ ) ⊗ Q . Let F/K be a finite Galois extension containing L , and write D = Gal( F/K ) , I ⊳ D for its inertia subgroup and
F = Frob − L/K .As functions from the Burnside ring of D to Q × / Q × , C v = c v ∼ D, I, e dim V F f ∼ D V (up to factors of 2 in Case (4ex) ).Proof. The N´eron model of A/ O K commutes with base change as A is semi-stable, so the minimal exterior form ω remains minimal in extensions of K ,and C v = c v .As V is unramified, D V = ( H det( | H | h , i| V H )) = D, I, ( | D | ef ) dim V F f det( h , i| V F f ) which is ∼ D, I, e dim V F f by Theorem 2.36f. This proves the last ∼ .As explained above, c v = D, I, (coker eN ) F f . So it remains to prove that D, I, φ ( e, f ) ∼
1, where φ ( e, f ) = (cid:12)(cid:12) (coker eN ) F f (cid:12)(cid:12) e − dim V F f ∈ Q × / Q × . Claim.
The function φ satisfies:(1) φ ( e, pf ) = φ ( e, f ) for p odd.(2) φ ( e, f ) = φ ( e, f ).(3) φ ( e,
2) = | coker N | .(4) φ (2 k e, f ) = φ (2 k , f ) for e odd.(5) φ ( e, f ) = 2 λ e,f φ (1 , f ) for some λ e,f ∈ Z .Before verifying the claim, let us use these properties to complete ourproof. Note that the asserted formula already holds up to multiples of 2 by(5) and Theorem 2.36f.Let W ⊳ I be the wild inertia subgroup. Then D, I, φ ( e, f ) (1,2) = D, I, n φ ( e, , | fφ ( e, , ∤ f o (3) = D, I, n | coker N | , | fφ ( e, , ∤ f o ∼ D, I, n , | fφ ( e, , ∤ f o ∼ I, I, φ ( e, ∼ I, W, φ ( ef, . If K has odd residue characteristic, then the wild inertia group W has oddorder and I, W, φ ( ef, = I, W, φ ( f, ∼ K has residue characteristic 2. Then [ I : W ] is odd, and I, W, φ ( ef, (4) = I, W, φ ( e, ∼ W, W, φ ( e, . If W is cyclic, this is ∼ W has no relations. If 2 ∤ [ L : K ],then F and F generate the same group, so W, W, φ ( e, = W, W, φ ( e, (3) = W, W, | coker N | ∼ . Otherwise we are in case (4ex) and we claim nothing about factors of 2. (cid:3)
EGULATOR CONSTANTS AND THE PARITY CONJECTURE 41
Proof of claim.
This is a purely module-theoretic statement about the G -modules X ( T ∗ ) ֒ → Hom( X ( T ) , Z ) and the monodromy pairing. Suppose G = h F i is a finite cyclic group, and N : M ′ ֒ → M is an inclusion of Z G -lattices of the same (finite) rank. Furthermore, suppose for every e ≥ G -invariant pairing h , i e : M/eM ′ × M/eM ′ −→ Q / Z . Then the function φ ( e, f ) = (cid:12)(cid:12) (coker eN ) F f (cid:12)(cid:12) e − rk M F f ∈ Q × / Q × satisfies (1)–(5):(1) As M ⊗ Q is a self-dual representation (every rational representationis self-dual), rk M F f ≡ rk M F pf mod 2. Write U = M/eM ′ and Z = F f . Then( m, n ) = h Zm − Z − m, n i e defines an alternating pairing U Z p × U Z p −→ Q / Z whose left kernel is U Z ∩ U Z p = U Z . So it is a perfect alternating pairing on U Z p /U Z , hencethis group has square order.(2) Again M ⊗ Q is self-dual, so rk M F f ≡ rk M F f mod 2. Next, the aboveformula for ( , ) with Z = F f defines an alternating pairing U F f × U F f → Q / Z whose left kernel is U Z ∩ U Z = U Z , so U Z /U Z has square order.(3) By (1) and (2), φ ( e,
2) = φ ( e, | G | ) = | M/eM ′ | e rk M = | M/M ′ | .(4) Replacing F by F f we may assume f = 1. We may also suppose that Fhas order a power of 2 by (1). It suffices to show that in the exact sequence0 −→ (cid:16) k M ′ k eM ′ (cid:17) F −→ (cid:16) M k eM ′ (cid:17) F −→ (cid:16) M k M ′ (cid:17) F −→ H (cid:16) G, k M ′ k eM ′ (cid:17) the first term has order e rk M F and the last one is 0. Because k M ′ k eM ′ ∼ = M ′ eM ′ ,it has odd order and hence has trivial H . Next, from the long exact coho-mology sequence for the multiplication by e map on M ′ we extract0 −→ M ′ F eM ′ F −→ (cid:16) M ′ eM ′ (cid:17) F −→ H ( G, M ′ )[ e ] . The last term is killed by | G | and e , and is therefore 0. So (cid:12)(cid:12)(cid:12)(cid:16) k M ′ k eM ′ (cid:17) F (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:16) M ′ eM ′ (cid:17) F (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) M ′ F eM ′ F (cid:12)(cid:12)(cid:12) = e rk M F , as required.(5) By (4), we only need to show that φ (2 k , f ) differs from φ (1 , f ) by apower of 2. But the first and the last term in the exact sequence (cid:16) M ′ k M ′ (cid:17) F f −→ (cid:16) MM ′ (cid:17) F f −→ (cid:16) M k M ′ (cid:17) F f −→ H (cid:16) h F f i , M ′ k M ′ (cid:17) are killed by 2 k , and the result follows from the definition of φ . (cid:3) Remark 3.29.
If we could also prove that φ (4 ,
1) = φ (2 ,
1) for φ as in 3.28,we would be able to deal with the exceptional case (4ex) of Theorem 3.2using an argument similar to that in § p = 2 in Theorem 1.6(b). (Embarass-ingly, this is purely a problem about Z C n -modules.)Combining Propositions 3.26 and 3.27 completes Case (4) of Proposition 3.10and the proof of Theorem 3.2.4. Applications to the parity conjecture
We now have a machine that, when supplied with a relation Θ betweenpermutation representations, confirms the p -parity conjecture for the twistsof A/K by the representations τ ∈ T Θ ,p coming from regulator constants.We turn to a class of Galois groups where these are enough to say somethingabout essentially all twists for some p .Specifically, we concentrate on Galois groups G = Gal( F/K ) that have anormal p -subgroup P . The type of results that we aim for is that knowing p -parity for all G/P -twists is sufficient to establish it for all G -twists. Inparticular, we prove Theorems 1.11 and 1.12.Apart from the machine itself (Theorem 1.6) the proofs rely only ongroup theory and basic parity properties of Selmer ranks and root numbers.Roughly speaking, we may consider any functions, such as τ w ( A/K, τ )or τ ( − h τ, X p ( A/F ) i that satisfy “self-duality” and “inductivity” as inProposition A.2(1,2). If two such functions agree on G/P -twists and on the τ ∈ T Θ ,p for those Θ that come from dihedral subquotients, this sometimesforces them to agree on all orthogonal G -twists, or at least on those twiststhat correspond to intermediate fields.We will not formulate the results of this section in this language. However,to be able in principle to extend them to a larger class of abelian varieties,we axiomatise the minimal compatibility requirements: Hypothesis 4.1 (Compatibility in dihedral subquotients) . Let
F/K be aGalois extension of number fields,
A/K an abelian variety and p a primenumber. We demand the following : whenever N ⊳ U are subgroups ofGal(
F/K ) with
U/N ∼ = D p n and τ = σ ⊕ ⊕ det σ for some 2-dimensionalrepresentation σ of U/N ,( − h τ, X p ( A/F N ) i U = w ( A/F U , τ ) . In other words, the p -parity conjecture holds for the twists by all such τ .(Recall that we regard C × C as a dihedral group as well.) For odd p , the hypothesis may be relaxed to subquotients U/N ∼ = D p ; this followsfrom the recent invariance result of Rohrlich (Prop. A.2(5) or [32] Thms. 1,2). EGULATOR CONSTANTS AND THE PARITY CONJECTURE 43
Theorem 4.2.
Hypothesis 4.1 holds for (1) (any p ) all elliptic curves over K whose primes of additive reductionabove 2 and 3 have cyclic decomposition groups in F/K (e.g. areunramified). (2) ( p = 2 ) all principally polarised abelian varieties over K whoseprimes of unstable reduction have cyclic decomposition groups in F/K (e.g. all semistable principally polarised abelian varieties). (3) ( p = 2 ) abelian varieties over K with a principal polarisation com-ing from a K -rational divisor, whose primes of unstable reductionhave cyclic decomposition groups in F/K , and with split semistablereduction at those primes above that have non-cyclic wild inertiagroups in F/K .Proof.
Apply Theorem 1.6 to the relations in Examples 2.21 and 2.22. (cid:3)
Throughout the section we implicitly use that X p behaves in an “´etale”fashion: for K ⊂ L ⊂ F an intermediate field, X p ( A/L ) = X p ( A/F ) Gal(
F/L ) (see e.g. [6] Lemma 4.14). We occasionally say that “ p -parity holds” for A/L or for a twist of A by τ , referring to Conjectures 1.2a, 1.2b.4.i. Parity over fields.Theorem 4.3.
Let A , p = 2 and F/K satisfy Hypothesis 4.1. Suppose
P ⊳
Gal(
F/K ) is a p -subgroup. If the p -parity conjecture holds for A overall subfields of F P /K , then it holds over all subfields of F/K .Proof.
Write G = Gal( F/K ) and V for Z ( P )[ p ], the p -elementary part ofthe centre of P . We may assume P = 1, so V is non-trivial. As V ischaracteristic in P , it is normal in G . We need to prove p -parity for A/F H for all subgroups H of G , and it holds when P ⊂ H by assumption. We useinduction on | G | to reduce G and H to small explicit groups. Thus, assumethe theorem holds for all proper subquotients | Gal( F ′ /K ′ ) | < | Gal(
F/K ) | .Fix H (cid:12) G . Suppose there is a subgroup 1 = U ⊳ G with U ⊂ P and HU = G . Applying the theorem to P/U ⊳
Gal( F U /K ), p -parity holds over allsubfields of F U /K , including the intermediate fields of F U /F HU . Applyingit again to U ⊳
Gal(
F/F HU ) shows that p -parity holds over the subfields of F/F HU , in particular F H .Hence we may assume that whenever U ⊳G is a subgroup of P , either U = 1or HU = G . In particular, HV = G as V ⊳ G is non-trivial. Furthermore, H ∩ V = 1 because it is normal in HV = G and H ( H ∩ V ) = H = G . Itfollows that G ∼ = H ⋉ V .Moreover, P = ( P ∩ H ) ⋉ V , as P contains V . The two constituentscommute, so this is a direct product and V = Z ( P )[ p ] = Z ( P ∩ H )[ p ] × V .So P ∩ H = 1, and hence P = V .Finally, we may assume that the action of H on V by conjugation isfaithful. Otherwise let W = ker( H → Aut V ) and note that W ⊳ H , so
W ⊳ HV = G . By induction, p -parity holds over all subfields of F W /K , inparticular over F H .We are reduced to the case G = H ⋉ F kp with H < GL k ( F p ) (an affine linear group) , where we need to show that p -parity for A/F H follows from p -parity overthe subfields of F F kp /K .The group G acts on the one-dimensional complex characters of F kp byconjugation. Let { χ i } be a set of representatives for the orbits, and let S i < G be the stabiliser of χ i . Extend χ i to a character ˜ χ i of S i by ˜ χ i ( hv ) = χ i ( v )for h ∈ S i ∩ H and v ∈ F kp . The representations Ind GS i ˜ χ i are irreducible anddistinct ([36] § C [ G/H ] ∼ = M i Ind GS i ˜ χ i . Indeed, both have dimension p k , so it is enough to check that each term onthe right is a consituent of C [ G/H ]; but h C [ G/H ] , Ind GS i ˜ χ i i = h H , Res H Ind GS i ˜ χ i i (Frobenius reciprocity) ≥ h H , Ind HS i ∩ H Res S i ∩ H ˜ χ i i (Mackey’s formula)= h H , Ind HS i ∩ H S i ∩ H i (definition of ˜ χ i )= h Res S i ∩ H H , S i ∩ H i = 1 (Frobenius reciprocity).Now considerΣ = (cid:8) i (cid:12)(cid:12) Ind GS i ˜ χ i self-dual (cid:9) = (cid:8) i (cid:12)(cid:12) χ ± i belong to the same H -orbit (cid:9) . For i ∈ Σ let M i consist of those elements of G that take χ i to χ ± i , and let ψ i = Ind M i S i ˜ χ i . Computing modulo 2, h H , X p ( A/F ) i = h C [ G/H ] , X p ( A/F ) i (Frobenius reciprocity) ≡ P i ∈ Σ h Ind GS i ˜ χ i , X p ( A/F ) i (Self-duality of X p ) ≡ P i ∈ Σ h ψ i , X p ( A/F ) i (Frobenius reciprocity) . The same computation for the root numbers (using A.2(1,2)) shows that w ( A/F H ) = Q i w ( A/F M i , ψ i ). So, it suffices to prove that( − h ψ i , X p ( A/F ) i = w ( A/F M i , ψ i ) . If χ i = , then S i = G , ψ i = and this p -parity holds by assumption.Otherwise, χ i = χ − i as p is odd, and ψ i factors through the D p -subquotient M i / ker ˜ χ i . In this case we know p -parity over its two bottom fields F S i and F M i , so it also holds for the twist of A/F M i by ψ i (Hypothesis 4.1). (cid:3) EGULATOR CONSTANTS AND THE PARITY CONJECTURE 45
Theorem 4.4.
Let
A, p = 2 and
F/K satisfy Hypothesis 4.1. Suppose thatthe Sylow 2-subgroup P of Gal(
F/K ) is normal. If the -parity conjectureholds for A over K and its quadratic extensions in F , then it holds over allsubfields of F/K .Proof.
Write G = Gal( F/K ) and pick H < G . We prove p -parity for A/F H . Step 1: Suppose G is a 2-group .There is a descending chain of subgroups G = U ⊃ . . . ⊃ U n = H withall inclusions of index 2. We show by induction that 2-parity holds overall quadratic extensions of F U i in F . For i = 1 this is true by assumption.Suppose this is true for i −
1, and let
L/F U i be a quadratic extension inside F .The Galois closure of the quartic extension L/F U i − has Galois group C ,C × C or D , as it is a 2-group. In all cases, 2-parity over quadraticextensions of F U i − implies 2-parity for all orthogonal twists of this Galoisgroup, in particular parity over L (for C see Corollary A.3(1,2); for D thisis Hypothesis 4.1.) Step 2: General case .As F H /F H ∩ P is Galois of odd degree, 2-parity for A/F H is equivalent tothat for A/F H ∩ P by Corollary A.3(3). Since P is a 2-group, by Step 1 itsuffices to establish 2-parity over F P and its quadratic extensions in F .Let Φ ⊳P be its Frattini subgroup, so P/ Φ ∼ = F k is its largest 2-elementaryquotient. As Φ is characteristic in P , it is normal in G , and F Φ /K is Galois.( F Φ is the compositum of all quadratic extensions of F P in F .) Replacing F by F Φ we may assume that Φ = 0 and P = F k , so by the Schur-Zassenhaustheorem G ∼ = U ⋉ F k with U of odd order.We want to prove 2-parity for all twists of A/F P by characters χ : F k → C × . Write L χ for F ker χ for such χ ; so [ L χ : F P ] ≤ F P /K is Galois of odd degree, 2-parity holds over L = F P , equiv-alently for the twist of A/F P by . More generally, G acts on charactersof F k by conjugation, and if χ = is G -invariant, then L χ /K is Galois withGalois group U × C . In this case, L χ is an odd degree Galois extension ofa quadratic extension of K in F , so again 2-parity holds over L χ and hencefor the twist of A/F P by χ .Now pick a general non-trivial χ = χ and let { χ i } ≤ i ≤ n be the completeset of its conjugates under G . The L i are conjugate fields, so the 2-parityconjecture for the twist by χ is equivalent to that for any of the χ i . Asthe orbit size n is odd, it suffices to check 2-parity for the twist of A/F P by ⊕ i χ i .Applying Hypothesis 4.1 in C × C -extensions of F P , 2-parity holds forthe twist by ⊕ φ ⊕ ψ ⊕ φψ for any characters φ, ψ of F k . Taking a sum of suchtwists shows that 2-parity for ⊕ i χ i is equivalent to 2-parity for ⊕ n ⊕ Q χ i .But this is a sum of G -invariant characters, for which 2-parity has alreadybeen established. (cid:3) Parity for twists.Theorem 4.5.
Let
A, p and
F/K satisfy Hypothesis 4.1. Assume that theSylow p -subgroup P of G = Gal( F/K ) is normal and G/P is abelian. If the p -parity conjecture holds for A over K and its quadratic extensions in F ,then it holds for all twists of A by orthogonal representations of G .Proof. Let τ be an orthogonal representation of G . By the analogue ofBrauer’s induction theorem for orthogonal representations [10] (2.1), τ = M i Ind GH i ρ ⊕ n i i for some H i < G , n i ∈ Z , and with ρ i either (a) trivial or (b) χ ⊕ ¯ χ with χ = ¯ χ one-dimensional or (c) a 2-dimensional irreducible that factors through adihedral quotient of H i .By inductivity (Corollary A.3(2)), it suffices to prove that( − h ρ i , X p ( A/F ) i Hi = w ( A/F H i , ρ i ) . We distinguish between the three possibilities for ρ i as above:Case (a). As G/P is abelian, its only irreducible self-dual representationsare those that factor through a C -quotient. By “self-duality” and “induc-tivity” (Corollary A.3(1,2)), the assumed parity over K and its quadraticextensions implies parity in all subfields of F P /K . By Theorems 4.3 and 4.4,it implies parity in all subfields of F/K , in particular for
A/F H i .Case (b). The formula holds by Corollary A.3(1).Case (c). Since the commutator of G is a p -group, the only dihedralsubquotients it has are D p k . By case (a), we know parity over F H i and itsquadratic extensions in F , so Hypothesis 4.1 implies parity for all irreducible2-dimensional representations of this subquotient. (cid:3) Remark 4.6.
For elliptic curves, the assumption that the p -parity conjec-ture holds for E over K and its quadratic extensions in F is known in anumber of cases. In particular ([1, 11, 14, 24, 25, 16, 6], [5, 3])(1) if K = Q ;(2) if E/K admits a rational p -isogeny, and for every prime v | p of K , • ( p > E is semistable, potentially multiplicative or potentiallyordinary at v , or acquires good supersingular reduction over anabelian extension of K v . • ( p = 3) E is semistable at v , • ( p = 2) E is semistable, and not supersingular at v .There are also results for modular abelian varieties over totally real fields[25, 17, 26] and a generalisation of (2) to abelian varieties with a suitable p g -isogeny [3]. EGULATOR CONSTANTS AND THE PARITY CONJECTURE 47
Remark 4.7.
The assumption in Theorem 4.5 that
G/P is abelian was onlyused to ensure that (a) p -parity holds in all intermediate fields of F P /K ,and (b) dihedral subquotients of G have the form D p n . So the theoremextends to other extensions that satisfy (a) and (b), e.g. G nilpotent with p = 2, or G/P ∼ = (odd) × (abelian 2-group) with p = 2. Example 4.8.
Let E/ Q be an elliptic curve, semistable at 2 and 3, andlet F = Q ( E [3]). We claim that the 3-parity conjecture holds for E overall subfields of F , and consequently over all subfields of F n = Q ( E [3 n ]) byTheorems 4.2 and 4.3.If either F/ Q is abelian or E/ Q admits a rational 3-isogeny, this is trueby [7] Thm. 1.2 and [5] Thm. 2 respectively. Otherwise, G = Gal( F/ Q ) isone of the following subgroups of GL ( F ):GL ( F ) , D or Sy = 2-Sylow of GL ( F ) . It is not hard to verify that in all three cases, the representations Ind GH H for subgroups H ⊂ C × C (these correspond to fields where E acquires a3-isogeny) and those with G/H abelian generate all orthogonal representa-tions. Again, as the 3-parity conjecture is known for
E/F H for such H , thisimplies 3-parity for all intermediate fields.The question whether 3-parity holds for all twists by self-dual represen-tations of Gal( F n / Q ) is more subtle, as we do not have an analogue of The-orem 4.5 in this case. In fact, suppose that G = Gal( F / Q ) ∼ = GL ( Z / Z ),i.e. as large as possible. Then there are precisely two irreducible orthogonalArtin representations τ , τ : G → GL ( C ) that can be realised over Q ( √ Q . It turns out that C Q Θ ( τ ⊕ τ ) = 1 for every G -relation Θ, sothe parity of h τ i , X ( E/F ) i cannot be computed from regulator constants.(It can be computed for all other C G -irreducible orthogonals.) Appendix A: Basic parity properties
For the convenience of the reader, we record a few basic facts related toroot numbers and the p -parity conjecture. Lemma A.1 (Determinant formula) . Let K be a local field and τ a contin-uous representation of the Weil group of K . Then w ( τ ⊕ τ ∗ ) = det( τ )( θ ( − , where θ is the local reciprocity map on K × . For an abelian variety A/K , w ( A, τ ⊕ τ ∗ ) = 1 . Proof.
For the first statement see [29] p.145 or [39] 3.4.7. The second is anelementary computation using [39] 3.4.7, 4.2.4. (cid:3)
Proposition A.2.
Let
F/K be a Galois extension of number fields, and
A/K an abelian variety. Write A = A ( F ) ⊗ C and X = X p ( A/F ) . For anArtin representation τ : Gal( F/K ) → GL n ( C ) , (1) (self-duality) h τ, A i = h τ ∗ , A i , h τ, X i = h τ ∗ , X i , w ( A, τ ) = w ( A, τ ∗ ) . (2) (inductivity) If K ⊂ L ⊂ F and τ = Ind Gal(
F/K )Gal(
F/L ) ρ then h τ, A i = h ρ, A i , h τ, X i = h ρ, X i , w ( A/K, τ ) = w ( A/L, ρ ) . (3) (odd degree base change) If F/K is Galois of odd degree, then rk(
A/K ) ≡ rk( A/F ) mod 2 , dim X p ( A/K ) ≡ dim X p ( A/F ) mod 2 ,w ( A/K ) = w ( A/F ) . (4) (orthogonality) If τ is symplectic, then h τ, A i is even and w ( A, τ ) = 1 . (5) (equivariance) If τ is self-dual and τ ′ is a Galois conjugate of τ (i.e.their characters are Galois conjugate), then h τ, A i = h τ ′ , A i , w ( A/K, τ ) = w ( A/K, τ ′ ) . Proof. (1) A is a rational representation, hence self-dual; X is self-dual aswell by [7] Thm. 1.1. The root number formula follows from Lemma A.1.(2) For A and X this is Frobenius reciprocity. The last formula is well-known; it is a consequence of inductivity in degree 0 of local ǫ -factors forWeil groups, [39] 4.2.4 and a simple determinant computation.(3) Follows from (1), (2) and the fact that the only self-dual irreduciblerepresentation of Gal( F/K ) is trivial.(4) A is a rational representation, so h τ, A i is even; w ( A, τ ) = 1 by [30]Prop. 8(iii) for elliptic curves, and by [33] Prop. 3.2.3 for abelian varieties.(5) The statement for A is clear, as it is a rational representation. That forroot numbers is clear from (4) in the symplectic case, and follows from [32]Thms 1,2 in the orthogonal case. (cid:3) Corollary A.3.
Suppose
F/K is a Galois extension, and
A/K an abelianvariety. Then the p -parity conjecture (1) holds for twists of A by representations of the form τ ⊕ τ ∗ . (2) holds for the twist of A/K by Ind
Gal(
F/K )Gal(
F/L ) ρ if and only if it holds forthe twist of A/L by ρ , if K ⊂ L ⊂ F . (3) holds for A/F if and only if it holds for
A/K , if [ F : K ] is odd. Acknowledgements.
We would like to thank the referee for carefully read-ing the manuscript and for helpful comments.
EGULATOR CONSTANTS AND THE PARITY CONJECTURE 49
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