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

TL;DR

This paper explores how Schur indices from representation theory can be used to identify families of Artin representations that impose divisibility conditions on the order of zeros of twisted L-functions of elliptic curves, under a conjectural framework.

Contribution

It introduces a novel application of Schur indices to predict divisibility properties of zeros of twisted L-functions and Selmer groups for elliptic curves, extending the Birch--Swinnerton-Dyer conjecture.

Findings

Existence of families of Artin representations with zero order divisible by p

Examples of twists by characters factoring through p-cyclotomic extension

Connections to Hilbert modular forms and Dirichlet characters

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.