Special value formula for the twisted triple product $L$-function and an application to the restricted $L^2$-norm problem

Abstract: We establish explicit Ichino's formulae for the central values of the triple product $L$-functions with emphasis on the calculations for the real place. The key ingredient for our computations is Proposition 6.8 which generalizes a result of Michel-Venkatesh. As an application we prove the optimal upper bound of a sum of restricted $L^2$-norms of the $L^2$-normalized newforms on certain quadratic extensions with prime level and bounded spectral parameter following the methods of Blomer.

• The paper derives explicit special value formulas for twisted triple product L-functions using a regularization technique at archimedean places.
• It extends Ichino’s formula to manage real local components, linking period integrals with automorphic forms.
• The formulation is applied to achieve optimal bounds on restricted L^2-norms of Maass newforms on quadratic extensions with prime level.

Special value formula for the twisted triple product $L$-function and the restricted $L^2$-norm problem

Introduction

The paper establishes special value formulae for central values of twisted triple product $L$-functions, specifically focusing on calculations at the real place. The work extends Ichino's formula using a regularization technique to manage archimedean local components and applies it to a problem of bounding restricted $L^2$-norms on quadratic extensions with prime level.

Special Value Formula for Triple Product $L$-function

The paper extends the work of Ichino by formulating explicit expressions for triple product $L$-functions over $\text{GL}_2$. It uses a regularization approach to handle local period integrals at archimedean places, extending previous non-archimedean results [Ichino 2008]. These formulas are crucial in understanding the connection between period integrals on quaternion algebras and automorphic forms.

Implementation Details

• Local Integrals: The calculation of local integrals at archimedean places uses a regularization process, generalizing previous results to include real components. This involves decomposing the Bessel function integral representations and utilizing the analytic properties of $K$-Bessel functions for convergence.
• Ichino's Formula Application: The use of Ichino's formula requires integrating automorphic forms over diagonal cycles in quaternion algebras, with normalization factors accounting for class numbers and discriminants to solve equation set across places.

Trade-offs

• The regularization approach offers more tractable computation for $L$-functions but requires careful handling of convergence issues, especially with complex places.
• Depending on dimension and symmetry of the form, we may prefer different lifting theorems or approximation schemes.

Application to the Restricted $L^2$-norm Problem

The paper applies these formulae to bounding the restricted $L^2$-norm for Maass forms on quadratic extensions, following methods similar to those in the level aspect proposed by Blomer [Blomer 2013].

Key Results

• Bounding the Norms: The paper demonstrates optimal bounds on sums of restricted $L^2$ norms of Maass newforms on extensions by leveraging orthogonality in triple products and intricate bounds achieved through $L$-functions.
• Level Aspect Approach: The work improves results in level aspect bounding by proving sharp bounds for newform norms through delicate analysis of Hecke eigenvalues truncated by Atkin-Lehner involutions.

Computational Considerations

• Zachary Blomer's method impacts the implementation heavily due to reliance on Weyl's law and the rapid decay properties of Bessel functions.
• Large-scale implementations may need optimization for handling series over basis of cusp forms especially with tensor product constraints from triple product nature.

Conclusion

The research provides a crucial extension of known computational methods by showing explicit value formulas for $L$-functions in the context of quaternion algebra automorphisms. Additionally, its application to bounding norms opens pathways for deeper exploration of arithmetic quantum chaos in modular symmetries. These results significantly contribute to the theory of automorphic forms and underscore the intersection of analytic and algebraic number theory.

Future Developments

Potential developments include refining bounds with optimized weight functions, extending results to higher rank groups, and exploring deeper geometric interpretations of these automorphic phenomena. Further collaborations on period integral models could extend these computational techniques to broader contexts in arithmetical algebraic geometry.