Periods of modular forms and applications to the conjectures of Oda and of Prasanna-Venkatesh

2025/07/07 by Xavier Guitart, Guitart, Xavier, Santiago Molina +1
Mathematics · #11F67 (primary) #11G40 (secondary) #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2507.05021

openalex publication_date 2025/07/07 · openalex created_date 2025/10/20 · openalex updated_date 2026/07/28

Abstract

We establish several formulas relating periods of modular forms on quaternion algebras over number fields to special values of L-functions. Our main inputs are the cohomological techniques for working with periods introduced in [Mol21], along with explicit versions of the Waldspurger formula due to Cai-Shu-Tian. We work in general even positive weights; when specialized to parallel weight 2, our formulas provide partial evidence for the conjectures of Oda and of Prasanna-Venkatesh in the case of forms associated to elliptic curves.

Citations

• Waldspurger formulas in higher cohomology
• Period Relations for Standard L-functions of Symplectic Type
• Eichler-Shimura isomorphism and group cohomology on arithmetic groups
• Automorphic cohomology, motivic cohomology, and the adjoint L-function
• On special zeros of p-adic L-functions of Hilbert modular forms

Related

• Period Relations for Quaternionic Elliptic Functions
• Special values of hypergeometric functions and periods of CM elliptic curves
• On certain period relations for cusp forms on GL_n
• Flach system on Quaternionic Hilbert--Blumenthal surfaces and distinguished periods
• On the quantitative variation of congruence ideals and integral periods of modular forms