# Hida families and p-adic triple product L-functions

## TL;DR

This paper constructs three-variable p-adic triple product L-functions for Hida families, establishes explicit interpolation formulas, and proves their factorization into anticyclotomic p-adic L-functions, advancing the understanding of special values and conjectures in number theory.

## Contribution

It introduces a new construction of p-adic triple product L-functions for Hida families and proves their explicit interpolation and factorization properties.

## Key findings

- Constructed three-variable p-adic triple product L-functions.
- Proved explicit interpolation formulas matching conjectural shapes.
- Established factorization into anticyclotomic p-adic L-functions.

## Abstract

We construct the three-variable p-adic triple product L-functions attached to Hida families of ellptic newforms and prove the explicit interpolation formulae at all critical specializations by establishing explicit Ichino's formulae for the trilinear period integrals of automorphic forms. Our formulae perfectly fit the conjectural shape of p-adic L-functions predicted by Coates and Perrin-Riou. As an application, we prove the factorization of certain unbalanced p-adic triple product L-functions into a product of anticyclotomic p-adic L-functions for modular forms. By this factorization, we give a new construction of the anticyclotomic p-adic L-functions for elliptic curves in the definite case via the diagonal cycle Euler system \'a la Darmon and Rotger and obtain a Greenberg-Stevens style proof of anticyclotomic exceptional zero conjecture for elliptic curves due to Bertolini and Darmon.