Four-variable $p$-adic triple product $L$-functions and the trivial zero conjecture

Abstract: We construct the four-variable primitive $p$-adic $L$-functions associated with the triple product of Hida families and prove the explicit interpolation formulae at all critical values in the balanced range. Our construction is to carry out the $p$-adic interpolation of Garrett's integral representation of triple product $L$-functions via the $p$-adic Rankin-Selberg convolution method. As an application, we obtain the cyclotomic $p$-adic $L$-function for the motive associated with the triple product of elliptic curves and prove the trivial zero conjecture for this motive.

• The paper constructs four-variable p-adic L-functions using Hida families and a p-adic Rankin-Selberg method, extending previous triple product formulations.
• The authors interpolate critical values to verify the trivial zero conjecture for motives associated with triple products of p-ordinary elliptic curves.
• Their approach leverages congruence and idempotent methods to yield insights into Galois representations underlying elliptic modular forms.

Four-variable $p$-adic triple product $L$-functions and the trivial zero conjecture

Introduction

The paper addresses the construction of four-variable $p$-adic $L$-functions associated with the triple product of Hida families and investigates their properties through interpolation formulae at all critical points in the balanced range. The authors leverage the $p$-adic Rankin-Selberg method to generalize Garrett's integral representation of triple product $L$-functions.

Galois Representations and Hida Families

The study involves the Galois representations associated with Hida families, a formulation relevant to the $p$-adic properties of elliptic modular forms. Each Hida family admits a big Galois representation, which plays a crucial role in understanding the arithmetic properties of the associated $p$-adic $L$-functions.

$p$-adic $L$-functions and Trivial Zeroes

A significant part of this study is focused on constructing $p$-adic $L$-functions that extend previous constructions of three-variable $p$-adic $L$-functions by including a cyclotomic variable. The authors prove the trivial zero conjecture for particular motives associated with the triple product of $p$-ordinary elliptic curves. They also tackle cases associated with the Greenberg-Benois trivial zero conjecture.

Key Results and Interpolation

The authors present the main result as follows: for a well-defined element in certain algebraic structures, properties are interpolated to give the behavior of $p$-adic $L$-functions at critical points, characterized by explicit conditions involving balanced regions and modular forms’ weights. This extends existing knowledge on $p$-adic properties by utilizing certain congruence and idempotent methods.

Applications to Elliptic Curves

An application is straightforward when considering the Galois representations associated with rank-eight motives derived from the triple product of elliptic curves. This segment resolves specific conjectures about trivial zeros, especially in scenarios when elliptic curves have split multiplicative reduction at $p$.

Conclusion

In conclusion, the paper constructs four-variable $p$-adic triple product $L$-functions with interpolations aligned with conjectural frameworks. These constructions accommodate existing theories and enrich the analytical arsenal for tackling trivial zero issues. The work lays a foundation for further research on $p$-adic properties in modular forms and elliptic curve products. Future advancements may involve extending these techniques to non-ordinary settings and more comprehensive families of forms.