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 $F$, denoted $L^{\mathrm{imp}}_p(\mathrm{Sym}^2 F)$. 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 $F$. It is in fact uniquely determined by its interpolation properties. We also prove that the function $L^{\mathrm{imp}}_p(\mathrm{Sym}^2 F)$ satisfies a functional equation. We use this $p$-adic $L$-function to prove a $p$-adic factorisation formula, expressing the geometric $p$-adic $L$-function attached to the self-convolution of $F$, as the product of $L^{\mathrm{imp}}_p(\mathrm{Sym}^2 F)$ and a Kubota-Leopoldt $L$-function. This extends a result of Dasgupta in the ordinary case. Using Beilinson-Flach classes constructed by Kings, Zerbes and the second author we construct motivic cohomology classes $b_f$, and prove that, under some hypotheses, they differ by a scalar factor from the higher cyclotomic classes constructed by Beilinson. Using this relation, we prove the interpolation formulae for $L^{\mathrm{imp}}_p(\mathrm{Sym}^2 F)$ and the factorisation formula.