Finite slope Triple Product p-adic L-functions over totally real fields

Abstract: We construct p-adic L-functions associated with triples of finite slope p-adic families of quaternionic automorphic eigenforms over totally real fields on Shimura curves. These results generalize a previous construction, joint work with D.Barrera, performed in the ordinary setting.

• The paper extends the construction of finite slope p-adic L-functions to quaternionic automorphic forms, eliminating earlier assumptions and broadening weight space applicability.
• It introduces a novel formal vector bundle framework to ensure the convergence of power series for nearly overconvergent modular forms.
• The work constructs a triple product p-adic L-function over totally real fields, offering new insights into arithmetic geometry and conjectures like Birch and Swinnerton-Dyer.

Finite slope Triple Product $p$-adic L-functions over Totally Real Fields

Introduction

The paper "Finite slope Triple Product p-adic L-functions over totally real fields" extends the construction of $p$-adic L-functions to accommodate finite slope $p$-adic families of quaternionic automorphic eigenforms over totally real fields on Shimura curves. Traditional constructions were confined to the ordinary case, but this work overcomes limitations regarding the iteration of the Gauss-Manin connection in a finite slope context by adapting techniques from Andreatta and Iovita.

Theoretical Framework

The study leverages the theory of formal vector bundles and introduces modifications suited to quaternionic setting over Shimura curves. The framework involves several mathematical structures such as quaternion algebras defined over totally real fields, maximal orders, and Shimura curves with moduli interpretations that extend to these contexts. These curves are parameterized in terms of quaternionic automorphic forms with applications drawn to the Birch and Swinnerton-Dyer conjecture and other related topics.

Main Results

• Extension of $\chi$-Iteration: The work extends the $\chi$-iteration of the Gauss-Manin connection to the universal character. This eliminates specific assumptions from previous works, broadening the applicability to the entire weight space.
• Formal Vector Bundles: A new type of formal vector bundle is defined, contributing to the $p$-adic interpolation of modular forms in the nearly overconvergent context. The utilization of these bundles ensures the convergence of associated power series.
• Triple product $p$-adic L-function: The paper constructs a triple product $p$-adic L-function associated with the quaternionic automorphic forms and proves its applicability to totally real fields, removing prior restrictions and improving universality and completeness.

Computational Techniques

Canonical groups and canonical subgroups are used to construct overconvergent modular sheaves that facilitate the $p$-adic interpolation. Moreover, the local analytic expansion of characters and their interaction with finite slope settings are crucial for deriving explicit arithmetic properties and ensuring computational feasibility.

Implications and Future Work

The construction of finite slope $L$-functions contributes to the broader understanding of automorphic forms and arithmetic geometry, with specific implications for the ongoing exploration of the Birch and Swinnerton-Dyer conjecture. The techniques and results facilitate potential future extensions to other algebraic structures or varied arithmetic contexts.

The methods could be adapted or extended to explore different types of modular forms or broader classes of automorphic forms, potentially influencing the development of new arithmetic conjectures or theorems. The theoretical advancements pave the way for further computational advancements in $p$-adic number theory and algebraic geometry.

Conclusion

The research provides a valuable extension of $p$-adic $L$-functions to finite slopes in the setting of quaternionic automorphic forms over Shimura curves, contributing significant theoretical advancements and practical methodologies for arithmetic applications over totally real fields.