Periods of modular forms and applications to the conjectures of Oda and of Prasanna-Venkatesh
Xavier Guitart, Santiago Molina
TL;DR
The paper develops a general framework that connects periods of modular forms on quaternion algebras to special values of L-functions over number fields. By combining cohomological period techniques with explicit Waldspurger-type formulas, it derives algebraicity results for central L-values across even weights and establishes period relations that specialize to elliptic curves in weight-2 cases. The work yields evidence for Oda’s conjecture and for Prasanna–Venkatesh’s conjectures, via Beilinson-type rationality statements for the relevant motives and explicit period computations, including adjoint L-values. It also extends the interaction between automorphic periods, modular symbols, and diagonal torus integrals to higher cohomology through PV action and higher Gross-type formulas, enriching the arithmetic understanding of L-values and Beilinson regulators in the setting of Hilbert and quaternionic automorphic forms.
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.