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

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.

arXiv commentBased on the first author's 2020 Warwick PhD thesis. Final version, to appear in "Ann. Math. Québec" (Henri Darmon birthday volume)

References in corpus (7)

• Rankin--Eisenstein classes and explicit reciprocity laws
• Euler systems for Rankin--Selberg convolutions of modular forms
• Decomposition de motifs abeliens
• Euler systems for Hilbert modular surfaces
• Iwasawa theory for Rankin--Selberg products of $p$-non-ordinary eigenforms
• Iwasawa theory for the symmetric square of a modular form
• Regulators for Rankin-Selberg products of modular forms