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arXiv · 1910.14128

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

Jonas Bergström·Neil Dummigan·David Farmer·Sally Koutsoliotas
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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.

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Jonas Bergström, Neil Dummigan, David Farmer, Sally Koutsoliotas. 2019-10-30. $\mathrm{GL}_2\times\mathrm{GSp}_2$ $L$-values and Hecke eigenvalue congruences. https://arxiv.org/abs/1910.14128

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