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

Alessandro Arlandini, David Loeffler

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.

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

Abstract

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

Paper Structure

This paper contains 45 sections, 46 theorems, 168 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Statement of the results
  3. Strategy of the proof
  4. L-functions and factorisations
  5. Notation and preliminaries
  6. Primitive and imprimitive L-functions
  7. Primitive and imprimitive factorisation
  8. p-adic L-functions
  9. Dirichlet characters
  10. P-adic Rankin--Selberg $L$-functions
  11. Cohomology and regulators
  12. Cohomology theories
  13. A complex diagram
  14. A p-adic diagram
  15. Cohomology of Kuga-Sato varieties
  16. ...and 30 more sections

Key Result

Theorem A

After possibly shrinking $U$, there exists a $2$-variable meromorphic $p$-adic $L$-function $L^{\mathrm{imp}}_p(\mathop{\mathrm{Sym}}\nolimits^2 F)\colon U\times\mathcal{W} \to \overline{\mathbb{Q}}_p$ with the following property: for all pairs $(t, j) \in \mathbb{N}^2$, with $t \in U \cap \mathbb{N where $f_t$ is the weight $t + 2$ specialization of $F$, and $(*)$ is an explicit factor, as long a

Figures (2)

  • Figure 1: Complex diagram
  • Figure 2: $p$-adic étale diagram

Theorems & Definitions (110)

  • Theorem A: Construction of a two-variable $\mathop{\mathrm{Sym}}\nolimits^2$ $p$-adic $L$-function
  • Theorem B: Dasgupta-style factorization formula
  • Theorem C: P-adic symmetric square functional equation
  • Definition 2.2.1
  • Definition 2.2.2
  • Remark
  • Definition 2.2.3
  • Proposition 2.2.4
  • proof
  • Definition 2.2.5
  • ...and 100 more