On the Birch-Swinnerton-Dyer conjecture and Schur indices

Matthew Bisatt,Vladimir Dokchitser

doi:10.1112/blms.12199

Matthew Bisatt, Vladimir Dokchitser

Open Access

https://doi.org/10.1112/blms.12199

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Abstract

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.

Highlights

For every odd prime p, we exhibit families of irreducible Artin representations τ with the property that for every elliptic curve E the order of the zero of the twisted L-function L(E, τ, s) at s = 1 must be a multiple of p
We give further examples where τ can 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
Our main tool is the theory of Schur indices from representation theory

Summary

Making the analytic rank divisible by p

Let τ be an irreducible faithful Artin representation of a Galois extension F/Q with Gal(F/Q) ∼= Cq Cpn non-abelian and with pn q−1. Assuming Conjecture 2.1, for any n such that pn q−1 and primitive character ψ of Gal(FpKpn /Kp) ∼= Cqpn−1 , we have ords=1 L(fE, ψ, s) ≡ 0 mod p, where Kpn is the nth layer of the p-cyclotomic extension of Q. We 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.

Birch–Swinnerton-Dyer conjecture and the Schur index

Schur indices in Cq Cpn