Hida families and p-adic triple product 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.

• The paper introduces explicit interpolation formulas for three-variable p-adic triple product L-functions linked to Hida families of elliptic newforms.
• It employs advanced Galois representations and Ichino-type local zeta integrals to match theoretical conjectures and enable factorization.
• The work has significant implications for arithmetic conjectures, including extensions of the Birch and Swinnerton–Dyer conjecture and studies of exceptional zero phenomena.

Hida Families and p-adic Triple Product L-functions

Introduction

The paper addresses the construction of three-variable $p$-adic triple product $L$-functions related to Hida families of elliptic newforms. It offers explicit interpolation formulas at critical specializations. These formulas correlate with conjectural forms postulated by Coates and Perrin-Riou and fit within the framework of $p$-adic $L$-functions for Hida families. A notable application includes the factorization of certain unbalanced $p$-adic triple product $L$-functions into products of anticyclotomic $p$-adic $L$-functions, which extends to the definite case for elliptic curves via the diagonal cycle Euler system.

Galois Representations and Triple Product $L$-Functions

For each primitive Hida family of elliptic modular forms, an associated Galois representation is defined, and central values of triple product $L$-functions are interpolated over the weight space. Given arithmetic points, the resulting $p$-adic Galois representations determine significant concurrency in the interpolation formulas, which are shown to mesh precisely with existing conjectures. The work extends Ichino’s formula concerning trilinear period integrals and applies it to the automorphic forms covering a broad array of modular forms in the context of $p$-adic variations.

Local and Global Perspectives

The paper explores local and global trilinear period integrals over $p$-adic fields and links them to $p$-adic $L$-functions. By utilizing the adequacy of local Whittaker newforms and Hecke operators, ichino-type local zeta integrals are defined for the $L$-function's Euler product. These integrals primarily emerge from intertwining operators in $p$-adic field representations and facilitate uniformity across arithmetic specializations.

Construction and Interpolation of $p$-adic $L$-Functions

The $p$-adic $L$-functions are shown to factor into products of pre-existing $p$-adic $L$-functions for the corresponding anticyclotomic modular forms. Furthermore, a comparison with complex $L$-functions reveals that interpolation formulas are intimately related to modular forms' central values, necessitating precise tuning of Euler factors and linking periods. The choice of local Galois representations plays a crucial role, especially in regards to the modification of archimedean and $p$-factor computations, which are essential for the interpolation process.

Applications and Arithmetic Conjectures

Conclusively, the paper posits broader implications for arithmetic algebraic geometry, notably in proving facets of the Birch and Swinnerton–Dyer conjecture in certain anticyclotomic settings. The factorization results provide a new method for studying exceptional zero conjectures when applied to elliptic curves. The implications are extended to a potential new proof of certain conjectures in $p$-adic number theory through alternative mappings and the formulation of improved $p$-adic $L$-functions.

Conclusion

Overall, this work establishes a comprehensive framework for $p$-adic triple product $L$-functions linked to Hida families, offering significant insights into both theoretical underpinnings and practical arithmetic applications. Its exploration into complex $L$-functions and $p$-adic analogues promises expansive future applications toward core conjectures in number theory.