Further refinement of strong multiplicity one for GL(2)

TL;DR

This paper sharpens the strong multiplicity one theorem for GL(2) automorphic representations, providing precise bounds on the number of places where their Hecke eigenvalues differ, and constructs examples to confirm the bounds' optimality.

Contribution

It offers a refined version of the strong multiplicity one theorem for non-dihedral automorphic representations of GL(2), with explicit bounds and sharpness demonstrations.

Findings

Established sharp lower bounds for differences in Hecke eigenvalues

Proved the bounds are optimal through explicit examples

Extended results to both general and non-dihedral cases

Abstract

We obtain a sharp refinement of the strong multiplicity one theorem for the case of unitary non-dihedral cuspidal automorphic representations for GL(2). Given two unitary cuspidal automorphic representations for GL(2) that are not twist-equivalent, we also find sharp lower bounds for the number of places where the Hecke eigenvalues are not equal, for both the general and non-dihedral cases. We then construct examples to demonstrate that these results are sharp.