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arXiv · 1708.02616

Non-abelian $p$-adic Rankin-Selberg $L$-functions and non-vanishing of central $L$-values

Fabian Januszewski
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Abstract

We prove new congruences between special values of Rankin-Selberg $L$-functions for $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$ over arbitrary number fields. This allows us to control the behavior of $p$-adic $L$-functions under Tate twists and to prove the existence of non-abelian $p$-adic $L$-functions for Hida families on $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$. As an application, we prove strong non-vanishing results for central $L$-values: We give sufficient local conditions for twisted central Rankin-Selberg $L$-values to be generically non-zero.

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Fabian Januszewski. 2017-08-08. Non-abelian $p$-adic Rankin-Selberg $L$-functions and non-vanishing of central $L$-values. https://arxiv.org/abs/1708.02616

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