Work

Theta cycles of modular forms modulo $p^2$

Year: 2026

Type: article

Abstract: The theta cycle of a modular form modulo a prime $p\geq 5$ is well understood. By contrast, the theta cycle modulo a power of p is still mysterious and experimentally erratic. Here, we completely determine the theta cycle of a weight $k < p$ modular form modulo $p^2$ on the initial segment of length p and we prove exact values or nontrivial bounds for the weight filtrations on $p-2$ further segments of length $p - k + 1$ . In particular, asymptotically as $p \rightarrow \infty ,$ we establish 50% of the theta cycle exactly, and we provide nontrivial bounds for 100% of it. We determine the first two low points exactly and $\left \lfloor \frac {p-k+1}{2} \right \rfloor $ further low points at regular positions. Moreover, we detect low points at exceptional positions which solve a quadratic equation modulo p , and which disturb the otherwise regular structure in the segments that we exhibit.

Source: Canadian Journal of Mathematics

DOI: 10.4153/s0008414x2610220x

Authors: Scott Ahlgren, Martin Raum, Olav K. Richter

Institutions: University of Illinois Urbana-Champaign, University of Illinois System, Chalmers University of Technology, University of Gothenburg, University of North Texas

Language: English

Topic: Advanced Algebra and Geometry

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,550

Funder: Vetenskapsrådet