On the Birch--Swinnerton-Dyer conjecture and Schur indices

For every odd prime $p$, we exhibit families of irreducible Artin representations $\tau$ with the property that for every elliptic curve $E$ the order of the zero of the twisted $L$-function $L(E,\tau,s)$ at $s\!=\!1$ must be a multiple~of~$p$. Analogously, the multiplicity of $\tau$ in the Selmer group of $E$ must also be divisible by $p$. We give further examples where $\tau$ can moreover be twisted by any character that factors through the $p$-cyclotomic extension, and examples where the $L$-functions are those of twists of certain Hilbert modular forms by Dirichlet charaters. These results are conjectural, and rely on a standard generalisation of the Birch--Swinnerton-Dyer conjecture. Our main tool is the theory of Schur indices from representation theory.


Making the analytic rank divisible by p
There is a standard "minimalist conjecture" that generically the L-function of an elliptic curve vanishes to order 0 or 1 at s = 1, depending on the sign in the functional equation. As we will illustrate, this has to be used with some caution: even when the associated Galois representation is irreducible, certain L-functions cannot vanish to order 1 at s = 1 -the order of their zero should be a multiple of a (possibly large) integer n.
More precisely, we look at twists of elliptic curves E by Artin representations τ and their L-functions L(E, τ, s), that is the L-function associated to the tensor product of τ with the Galois representation of E. When τ factors through F/Q this is a factor of L(E/F, s), much like the Artin L-function L(τ, s) is a factor of the Dedekind ζ-function of F . Throughout the article p and q will be distinct odd primes. We write , for the usual inner product of characters of representations of finite groups (embedding them into C if necessary): thus X, τ is the multiplicity of τ in X if τ is irreducible.
Theorem 1.1. Let E/Q be an elliptic curve. Let τ be an irreducible faithful Artin representation of a Galois extension F/Q with Gal(F/Q) ∼ = C q ⋊ C p n non-abelian and with p n ∤ q−1. (i) If the Birch-Swinnerton-Dyer conjecture for Artin twists (Conjecture 2.1) holds, then ord s=1 L(E, τ, s) ≡ 0 mod p. (ii) If the ℓ-primary part of the Tate-Shafarevich group X(E/F )[ℓ ∞ ] is finite, then where ℓ is any prime and X ℓ (E/F ) is the Pontryagin dual of the ℓ ∞ -Selmer group of E/F tensored with Q ℓ , viewed as a representation of Gal(F/Q).
This result follows from Theorem 2.5 and Theorem 3.2(iii). The main question we would like to raise, of course, is whether this behaviour of L-functions or Selmer groups can be explained without appealing to the conjectures.
It is reasonably straightforward to construct such Galois extensions F/Q. Consider for simplicity the case when C p n acts on C q through C p . Such fields F = F p n are constructed as the compositum of a C p n -extension K p n /Q and an extension F p /Q with Galois group C q ⋊ C p that shares a common degree p subfield K p with K p n . The irreducible faithful Artin representations of Gal(F/Q) are all of the form τ ⊗ χ, for any irreducible p-dimensional representation of Gal(F p /Q) and any 1-dimensional representation χ of Gal(K p n /Q) of order p n (see Proposition 3.1).
Q For example, K p n could be the n th layer of the p-cyclotomic tower of Q, that is the unique degree p n subfield of Q(ζ p n+1 ), where ζ p n+1 is a primitive p n+1 -th root of unity. This gives the following: Corollary 1.2. Suppose that F p /Q is Galois with Gal(F p /Q) ∼ = C q ⋊ C p nonabelian, and that its degree p subfield K p is the first layer of the p-cyclotomic extension of Q. Let E/Q be an elliptic curve and τ an irreducible faithful representation of Gal(F p /Q). If Conjecture 2.1 holds, then for all finite order characters χ that factor through the p-cyclotomic extension with χ q−1 = 1, If τ is a representation of Gal(F/Q) such that τ = Ind K/Q ψ for some subfield K ⊂ F , then we have an equality of L-functions L(E, τ, s) = L(E/K, ψ, s) for any elliptic curve E/Q. In our setup, all irreducible faithful representations τ are induced from characters. More concretely, if Gal(F p n /Q) ∼ = C q ⋊C p n is non-abelian, such that C p n acts on C q through C p , then τ = Ind Kp/Q ψ where K p is the degree p subfield of F p n and ψ is a primitive character of order qp n−1 . In particular, we get the following consequence for L-functions of certain modular forms. Corollary 1.3. Suppose that F p /Q is Galois with Gal(F p /Q) ∼ = C q ⋊ C p nonabelian, and that its degree p subfield K p is the first layer of the p-cyclotomic extension of Q. Let E/Q be an elliptic curve, let f E be the modular form attached to E and let f E be the Hilbert modular form which is the base-change of f E to the (totally real cyclic) extension K p /Q. Assuming Conjecture 2.1, for any n such that p n ∤ q−1 and primitive character ψ of Gal(F p K p n /K p ) ∼ = C qp n−1 , we have where K p n is the n th layer of the p-cyclotomic extension of Q.
Question 1.4. Our approach relies on elliptic curves. Are there similar phenomena for modular forms that do not correspond to elliptic curves? Example 1.5. As a concrete example, take p = 3 and q = 7. For the degree 7 non-Galois extension F 1 (see diagram above) take the field F 1 = Q(α) of discriminant 3 8 7 12 , where α is a root of x 7 −42x 5 −70x 4 +168x 3 +126x 2 −84x−45. As in the above discussion, take K 3 n = Q(ζ 3 n+1 ) + and set F 3 n = F 1 K 3 n , the n th layer of the p-cyclotomic tower of F 1 . The field F 3 is the Galois closure of F 1 and Gal(F 3 /Q) ∼ = C 7 ⋊ C 3 non-abelian; this group is an analogue of a dihedral group with C 2 replaced by C 3 .
The group Gal(F 3 /Q) ∼ = C 7 ⋊ C 3 has three 1-dimensional representations that come from the C 3 -quotient, and two 3-dimensional irreducible representations τ 0 , τ ′ 0 , which are induced from 1-dimensional characters ψ 0 , ψ ′ 0 of C 7 . The irreducible representations of Gal(F 3 n /Q) ∼ = C 7 ⋊ C 3 n are the 1-dimensional representations lifted from the C 3 n -quotient, and 3-dimensional irreducibles that can all be written as τ = τ 0 ⊗ χ or τ = τ ′ 0 ⊗ χ for some 1-dimensional χ; note that these can therefore also be expressed as The faithful ones are precisely the ones with χ of maximal order, equivalently with ψ of order 7 × 3 n−1 .
Now let E/Q be an elliptic curve. The L-function in Theorem 1.1 can be expressed in several ways: if, say, where f E is as in Corollary 1.3. In this setting, our prediction is that the order of vanishing of this L-function is necessarily a multiple of 3, so long as τ does not factor through C 7 ⋊C 3 (equivalently if the order of χ is at least 9). As we will explain in §2-3, the corresponding statement is provably true for the Mordell-Weil group E(F 3 n ), which is how we obtain the prediction for L-functions and Selmer groups.
Finally, let us note that it is possible to make a prediction for analytic ranks that do not involve twisted L-functions, although it becomes a little cumbersome. Using the subfield lattice of F 3 n /Q and inductivity of L-functions, one checks that Observe that the faithful representations τ : Gal(F 3 n /Q) → GL 3 (Q) have Galois conjugate images, since they are induced from Galois conjugate 1-dimensional ψ's. Thus, if we assume Conjecture 2.1 or Deligne's conjecture on Galois-equivariance of L-values [Del79, Conjecture 2.7ii], the orders of vanishing of their L-functions should all be equal, and hence the order of vanishing of the right-hand term in the above equation is a multiple of 3 × 3 × (7−1)(3 n −3 n−1 ) 3 2 = 4 × 3 n . In particular, if the L-values at s = 1 are non-zero for E/F 3 n−1 and E/K 3 n (and hence for E/K 3 n−1 ), then the order of the zero of L(E/F 3 n , s) must be a multiple of 4 × 3 n . More generally, the same technique yields the following result.
Corollary 1.6. Let F/Q be a Galois extension with Gal(F/Q) ∼ = C q ⋊ C p n nonabelian, where the image of C p n in Aut C q has order p r and p n ∤ q−1. Suppose E/Q is an elliptic curve such that L(E/K, 1) = 0 for all proper subfields K F . If Conjecture 2.1 holds, then Remark 1.7. At present we do not have examples where the orders of vanishing of such L-functions are non-zero, as their conductors appear to be too large for any extensive numerical search. We also cannot guarantee a zero at s = 1 by forcing the L-function to be essentially antisymmetric about that point: the twisting Artin representations τ (or τ ⊗ χ) above are never self-dual, so the functional equation relates L(E, τ ) to L(E, τ * ) and the root number ("sign") cannot be used to force a zero. The latter is a general feature of our approach, see Remark 2.7. Remark 1.8. As will be clear from §2-3, Theorem 1.1 applies generally to abelian varieties over number fields, rather than elliptic curves over Q.
Remark 1.9. The Galois representation H 1 et (E, Q ℓ ) C ⊗ τ can be irreducible, so the multiplicity of the order of vanishing is not explained by a decomposition of the Galois representation. Moreover, the L-series is not the (formal) p th power of another L-series. For example, if G = C 7 ⋊ C 9 and v is a prime of good reduction of E such that Frobenius at v is an element of order 7 in G, then the Euler factor at v is , which is visibly not a cube; here α and β are the Frobenius eigenvalues at v of E, and ζ 7 a suitable primitive 7-th root of unity.

Birch-Swinnerton-Dyer conjecture and the Schur index
Statements that concern the Birch-Swinnerton-Dyer conjecture usually suppose properties about a given L-function so as to ascertain information about the rank (e.g. Coates-Wiles, Gross-Zagier, Kolyvagin). Our approach is somewhat peculiar: we are traversing the opposite direction by using the Mordell-Weil group to derive a feature of the L-function. We rely on the following generalisation of the Birch-Swinnerton-Dyer conjecture.
Conjecture 2.1 (Birch-Swinnerton-Dyer, Deligne-Gross; see [Roh90] p.127). Let A be an abelian variety over a number field K, and let τ be a representation of Gal(F/K) for some finite Galois extension F/K. Then L(A/K, τ, s) has analytic continuation to C and The key observation is that since the Galois group acts on a Z-lattice, A(F ) C is a rational representation. Therefore certain complex irreducible representations τ cannot appear with multiplicity 1 in A(F ) C ; this aspect is measured by the Schur index m Q (τ ). In contrast, the analogous property is not obvious (and unknown in general) for either the L-function of an abelian variety or the Q ℓ -representation on the dual Selmer group X ℓ (A/F ).
Definition 2.2. Let G be a finite group and F a subfield of C. We say a complex representation τ of G is realisable over F if it is conjugate to a representation that factors as G → GL n (F ) ⊂ GL n (C) for some n. The Schur index m F (τ ) is the maximal integer m such that for all representations σ of G that are realisable over F , the multiplicity τ, σ is a multiple of m.
Example 2.3. The Schur index m Q (τ ) of the 2-dimensional irreducible representation τ of the quaternion group Q 8 is 2. Hence τ , despite having rational trace, cannot be realised by matrices in GL 2 (Q); however τ ⊕ τ is realisable in GL 4 (Q).
Remark 2.4. Note that for any field F , m F (τ ) ≤ dim τ as the regular representation is realisable over Q. In fact m F (τ ) always divides the dimension dim τ , see e.g. [Isa76, Corollary 10.2].
Theorem 2.5. Let F/K be a Galois extension of number fields, and let τ be an irreducible Artin representation of Gal(F/K). Then for all abelian varieties A/K, the multiplicity of τ in A(F ) C is divisible by m Q (τ ). In addition: Proof. By construction, A(F ) C is realisable over Q so by definition m Q (τ ) divides A(F ) C , τ . The L-function statement now follows directly from Conjecture 2.1.
from which the second part follows.
Remark 2.6. Without the finiteness assumption on X, the dual Selmer group X ℓ (A/F ) is not known to be a rational or even an orthogonal representation of the Galois group (although it is known to be self-dual, see [DD09]). Thus, as the ℓ-adic Schur index m Q ℓ (τ ) can be 1, there is no obvious representation-theoretic reason for the multiplicity of τ in X ℓ (A/F ) to be a multiple of m Q (τ ); see Theorem 3.2 for an example of such a τ .
Remark 2.7. The reason for the restriction on the order of vanishing of the Lfunction is fairly well-understood for self-dual representations τ with Schur index 2 (for example the quaternion representation in Example 2.3). In this case the conjectural functional equation is of the form L(A, τ, s) = ±L(A, τ, 2 − s) × (Γ-factors and exponential). So the parity of the order of vanishing at s = 1 is determined by the sign ±, which is given by the global root number W (A, τ ) and known to be + whenever τ is symplectic and in many cases when τ is orthogonal with Schur index 2, see [Roh96, Proposition 2] and [Sab07, Theorem 0.1].
It is tempting to use the sign in the functional equation to force a zero of the L-function for a representation τ with large Schur index m = m Q (τ ). If Conjecture 2.1 is true, the order of vanishing is a fortiori at least m. Curiously enough, this is impossible to achieve: if m > 2, the representation τ cannot be self-dual by the Brauer-Speiser theorem. Thus the functional equation relates L(A, τ, s) to L(A, τ * , 2 − s), and the root number cannot be used to force the L-function to vanish at s = 1.

Schur indices in C q ⋊ C p n
We now compute the Schur indices of representations of C q ⋊ C p n appearing in Theorem 1.1. We only prove that the Schur index is divisible by p without determining it exactly, so the bounds on orders of vanishing of L-functions that we have given may be suboptimal. For example, if τ is an irreducible faithful representation of C 19 ⋊ C 3 4 (with the largest possible action), then m Q (τ ) = 9.
For a field F and representation τ , we let F (τ ) denote the finite abelian extension of F generated by the values of the trace of τ . We further let ζ m denote a primitive m th root of unity and N F /K be the norm map for any field extension F /K. Proposition 3.1. Let p, q be distinct odd primes and G = C q ⋊ C p n , where the image of C p n in Aut C q has order p r . Let τ be a complex irreducible representation of G. Write X = C q × C p n−r ⊳ G. (i) If τ is unfaithful then τ is lifted either from C p n or from C q ⋊ C p n−1 .
(ii) If τ is faithful, then dim τ = p r and there is a faithful 1-dimensional representation of X such that τ = Ind G X ψ. Conversely, the induction of a faithful 1dimensional representation ψ of X gives a faithful irreducible representation of G.
(iii) Every faithful irreducible representation τ of G may be written as τ r ⊗ χ for some faithful irreducible representation τ r of C q ⋊ C p r and faithful 1-dimensional representation χ of C p n . (iv) If τ = Ind G X ψ is faithful and F ⊂ C is a field, then F (ψ) = F (ζ p n−r , ζ q ) and Proof. The group G has presentation G = a, b | a q = b p n = id, bab −1 = a j where j has order p r modulo q. The subgroup X is a, b p r ; it is the centraliser of C q . For a representation ψ of X and a element g ∈ G we write g ψ for the conjugate representation defined by g ψ(h) = ψ(ghg −1 ).
(i) The maximal quotients of G are C p n and (if r < n) C q ⋊ C p n−1 , so if τ is not faithful, it factors through one of these.
(ii) By [Ser77, Proposition 25], every faithful representation of G is necessarily induced from a 1-dimensional representation ψ of X; in particular dim τ = p r . Moreover, since ker ψ is normal in G (as X is normal in G and ker ψ is characteristic in the cyclic group X), we have ker ψ ⊆ ker τ , and hence ψ must be faithful.
Conversely, h → b k hb −k are distinct automorphisms of X for 0 ≤ k < p r − 1, so if ψ is a faithful 1-dimensional representation of X, then ψ, b ψ, . . . , b p r −1 ψ are all distinct. Thus τ, τ = ψ, Res G X Ind G X ψ = ψ, 0≤k<p r b k ψ = 1 by Frobenius reciprocity and Mackey's formula, and so τ is irreducible. It is clearly faithful by (i).
(iii) Let τ = Ind G X ψ, for some faithful 1-dimensional ψ of order qp n−r . We can rewrite this as ψ = ψ q ⊗ ψ p n−r where ψ m has order m. Now τ r = Ind G X ψ q is the inflation of a faithful representation of C q ⋊ C p r . Let χ be a 1-dimensional representation of G which factors through C p n such that Res G X χ = ψ p n−r . The push-pull formula shows that τ = τ r ⊗ χ, as claimed.
(iv) If τ is faithful, then by (ii) ψ is a faithful 1-dimensional representation of X ∼ = C qp n−r , hence F (ψ) = F (ζ qp n−r ). To compute F (τ ), it suffices to compute the induced character on the conjugacy classes of G which have nonempty intersection with X. Since X ⊳ G, it follows that F (τ ) = F (Res G X τ ). Note that b p r is central in G and τ is irreducible so τ (b p r ) must be scalar by Schur's lemma; as Res G X τ contains ψ as a constituent, this scalar is multiplicationby-ζ p n−r , hence ζ p n−r ∈ F (τ ). For a x b p r y ∈ X we have tr τ (a x b p r y ) = ζ y p n−r tr τ (a x ), so F (τ ) is generated over F by ζ p n−r and the traces tr τ (a x )) for 1 ≤ x ≤ q.
As in the proof in (ii), Res G X τ = 0≤k<p r b k ψ, so tr τ (a x ) = t∈H ζ xt q , where H is the unique index subgroup of order p r contained in (Z/qZ) × . Note that for any polynomial f ∈ Q[X], f (ζ q ) is Gal(Q/Q)-conjugate to f (ζ x q ) whenever q ∤ x, and hence f (ζ x q ) ∈ Q(f (ζ q )) since Q(f (ζ q ))/Q is abelian. In particular, letting f (X) = t∈H X t (where we fix representatives for H), we see that t∈H ζ xt q ∈ Q( t∈H ζ t q ) for all x. Hence F (τ ) = F (ζ p n−r , t∈H ζ t q ) as claimed.
(v) First note that X is normal, abelian and equal to its own centraliser, X = C G (X), as otherwise b k ∈ C G (X) for some k with p r ∤ k which doesn't commute with a. Since by assumption the (abelian) extension F (ψ)/F (τ ) has degree p r , the representation ψ must have p r distinct Gal(F (ψ)/F (τ ))-conjugates, which then must be precisely the constituents of Res G X τ . Thus (G, X, τ ) is an F -triple, in the terminology of [Isa76, Definition 10.5]. Noting that G = XC p n , it then follows from [Isa76,Theorem 10.10] that m F (τ ) = 1 if and only if ζ p n−r ∈ N F (ψ)/F (τ ) F (ψ).
Theorem 3.2. Let p, q be distinct odd primes and G = C q ⋊ C p n , where the image of C p n in Aut C q has order p r and 0 < r ≤ n. Let τ be a complex irreducible faithful representation of G. Then: (i) The Schur index m Q (τ ) = p s for some 0 < s ≤ r if p n ∤ q−1, and is 1 otherwise; (ii) The Schur index m Qq (τ ) = m Q (τ ); (iii) The Schur index m Q ℓ (τ ) = 1 for every prime ℓ = q.