Modular Periods, Motives, and Single-Valued Anomalies: Extending the Gross-Zagier Paradigm

Brown and Fonseca (2025) provide a geometric/motivic interpretation of the Gross-Zagier conjecture, connecting modular periods and special values of L-functions. This idea proposes to search for and study settings where the period map or single-valued period constructions exhibit anomalies—for example, in higher-rank motives, non-congruence modular forms, or settings with unusual symmetry. By developing a theory of “single-valued period anomalies,” we could explain or predict where traditional conjectures like Gross-Zagier break down or require modification. This project is novel in its synthesis: it brings together recent motivic techniques, period integrals, and the detection of unexpected or unexplained arithmetic phenomena, opening new avenues in the theory of motives and special values.

References:

1. Single-valued periods of meromorphic modular forms and a motivic interpretation of the Gross-Zagier conjecture. Francis Brown, Tiago J. Fonseca (2025).