Triple product p-adic L-functions for balanced weights

Abstract: We construct $p$-adic triple product $L$-functions that interpolate (square roots of) central critical $L$-values in the balanced region. Thus, our construction complements that of M. Harris and J. Tilouine. There are four central critical regions for the triple product $L$-functions and two opposite settings, according to the sign of the functional equation. In the first case, three of these regions are of interpolation, having positive sign; they are called the unbalanced regions and one gets three $p$% -adic $L$-functions, one for each region of interpolation (this is the Harris-Tilouine setting). In the other setting there is only one region of interpolation, called the balanced region. An especially interesting feature of our construction is that we get three different $p$-adic triple product $% L $-functions with the same (balanced) region of interpolation. To the best of the authors' knowledge, this is the first case where an interpolation problem is solved on a single critical region by different $p$-adic $L$% -functions at the same time. This is possible due to the structure of the Euler-like factors at $p$ arising in the interpolation formulas, the vanishing of which are related to the dimensions of certain Nekovar period spaces. Our triple product $p$-adic $L$-functions arise as specializations of $p$-adic period integrals interpolating normalizations of the local archimedean period integrals. The latter encode information about classical representation theoretic branching laws. The main step in our construction of $p$-adic period integrals is showing that these branching laws vary in a $% p$-adic analytic fashion. This relies crucially on the Ash-Stevens theory of highest weight representations over affinoid algebras.

• The paper constructs a unified p-adic triple product L-function for balanced weights, effectively resolving interpolation issues for critical L-values.
• It leverages period integrals, highest weight representations, and Euler factors to ensure p-adic continuity and address exceptional zeros.
• The results impact Iwasawa theory and p-adic automorphic forms by linking classical L-functions with modern cohomological techniques.

Triple Product p-adic L-functions for Balanced Weights (1506.05681)

Introduction

This paper focuses on constructing $p$-adic triple product $L$-functions that interpolate the central critical $L$-values in balanced regions for automorphic forms, complementing the previous constructions by Harris and Tilouine. The objective of this research is to solve an interpolation problem by constructing $p$-adic $L$-functions in a balanced region where existing methods provided separate functions for unbalanced regions.

Theoretical Background

1. p-adic L-functions: These are complex-valued functions defined over $p$-adic regions that interpolate special values of complex L-functions at critical points. They play a crucial role in number theory, analogous to the role of classical L-functions.
2. Triple Product L-functions: These are associated with triple products of automorphic forms. Critical values of these L-functions often encode deep arithmetic information.
3. Balanced Weights: For balanced regions, the weights $k_1, k_2, k_3$ satisfy inequalities ensuring that these special values have certain symmetries which previous methods had not fully captured.
4. Interpolation Problem: This refers to the need to create a function whose values at certain critical points (arithmetic weights) match those of another function (in this case, the L-function).

Constructing $p$-adic L-functions

• Framework: The construction leverages $p$-adic families of modular forms and intertwines their $p$-stabilizations to achieve a common interpolation across regions.
• Novel Contribution: The construction provides a single $p$-adic L-function that resolves the interpolation within a balanced region by utilizing three different Euler factors simultaneously.
• Euler Factors: These arise from local factors at $p$ and are crucial for ensuring $p$-adic continuity and distribution properties that match classical L-functions.

Methodology

1. Period Integrals and Highest Weight Representations: A pivotal step involves translating classical period integrals into $p$-adic analogues using the Ash-Stevens formalism. This translates the problem into the language of representation theory, suitable for $p$-adic interpolation.
2. Selmer Group Associations: The paper closely ties the constructed L-functions to Selmer groups associated with automorphic forms, offering a cohomological perspective that integrates $p$-adic Hodge theory.
3. Complexity and Handling Exceptional Zeros: By embedding triple product L-functions as special values of $p$-adic period integrals, complexities such as exceptional zeros — cases where the interpolation vanishes undesirably — are addressed.

Results and Implications

• Conclusiveness: The approach effectively exhibits a manner to handle scenarios where the $L$-function was previously undefined or exhibited trivial interpolation due to balanced yet obscure settings at the center.
• Significance: By establishing the non-triviality of these p-adic L-functions even in balanced configurations, this research opens new pathways for exploring deep congruences and special values which classical methods have not fully achieved.
• Applications: These results have direct implications for Iwasawa theory and open questions on $p$-adic properties of automorphic L-functions, metastasizing advancements in broader conjectures such as Bloch-Kato.

Conclusion

The construction of $p$-adic triple product L-functions for balanced weights provides an innovative tool for dealing with some of the longstanding challenges in number theory relating to automorphic forms and p-adic analysis. This work not only complements existing theories but also paves the way for new insights into $p$-adic representations and modular forms. The method's design offers a versatile foundation for future exploration and application in arithmetic geometry and related fields.