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arXiv · 2103.16380

On the factorisation of the $p$-adic Rankin-Selberg $L$-function in the supersingular case

Alessandro Arlandini·David Loeffler
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Abstract

Given a cusp form $f$ which is supersingular at a fixed prime $p$ away from the level, and a Coleman family $F$ through one of its $p$-stabilisations, we construct a $2$-variable meromorphic $p$-adic $L$-function for the symmetric square of $F$. We prove that this new $p$-adic $L$-function interpolates values of complex imprimitive symmetric square $L$-functions, for the various specialisations of the family $F$. We use this $p$-adic $L$-function to prove a $p$-adic factorisation formula, expressing the geometric $p$-adic $L$-function attached to the Rankin--Selberg convolution of $f$ with itself as a the product of the $p$-adic symmetric square $L$-function of $f$ and a Kubota-Leopoldt $L$-function. This extends a result of Dasgupta in the ordinary case.

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Alessandro Arlandini, David Loeffler. 2021-03-30. On the factorisation of the $p$-adic Rankin-Selberg $L$-function in the supersingular case. https://arxiv.org/abs/2103.16380

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