$\mathrm{GL}_2\times\mathrm{GSp}_2$ $L$-values and Hecke eigenvalue congruences

TL;DR

This paper explores congruences of Hecke eigenvalues between automorphic representations on certain groups, linking them to prime divisors of special tensor-product $L$-values through computational techniques.

Contribution

It provides experimental evidence of Hecke eigenvalue congruences related to algebraic parts of tensor-product $L$-values for specific automorphic groups.

Findings

Identified congruences between automorphic representations of GSp2 and other groups.

Computed approximations of $L$-values to detect prime divisors.

Linked congruences to algebraic parts of critical $L$-values.

Abstract

We find experimental examples of congruences of Hecke eigenvalues between automorphic representations of groups such as $\mathrm{GSp}_2(\mathbb{A})$, $\mathrm{SO}(4,3)(\mathbb{\mathbb{A}})$ and $\mathrm{SO}(5,4)(\mathbb{A})$, where the prime modulus should, for various reasons, appear in the algebraic part of a critical "tensor-product" $L$-value associated to cuspidal automorphic representations of $\mathrm{GL}_2(\mathbb{A})$ and $\mathrm{GSp}_2(\mathbb{A})$. Using special techniques for evaluating $L$-functions with few known coefficients, we compute sufficiently good approximations to detect the anticipated prime divisors.