New Types of Sturm bounds via $p$-adic transfer methods

TL;DR

This paper introduces a novel p-adic transfer method to extend Sturm bounds from classical modular forms to quasimodular forms, enabling uniform bounds for forms with integer or modular coefficients and opening avenues for broader applications.

Contribution

The paper develops a new p-adic transfer technique to lift Sturm bounds between different modular form spaces using non-geometric inputs, extending bounds to quasimodular forms.

Findings

Transferred Sturm bounds to quasimodular forms of level one.

Established uniform bounds for forms with integer or modular coefficients.

Potential for applying the method to other modular and non-modular objects.

Abstract

Sturm's theorem states that a modular form with coefficients in $\mathbb{Z}$ or $\mathbb{Z}/m\mathbb{Z}$ can only have an explicitly bounded order of vanishing at infinity. This result is one of the most powerful computational tools in the study of modular forms, and has widespread applications to congruences and other kinds of explicit calculations in mathematics and physics. In this paper, we formulate a new ``$p$-adic transfer method" that lifts Sturm-type bounds from one space to another using exclusively non-geometric inputs. As an application, we transfer the Sturm bounds for classical modular forms to the space of quasimodular forms of level one. These bounds are applicable uniformly for quasimodular forms with coefficients in $\mathbb{Z}$ or $\mathbb{Z}/m\mathbb{Z}$, which extends the non-uniform results for $\mathbb{Z}/m\mathbb{Z}$ only which can be derived from classical theories. We also discuss the potential for future applications to other quasi- and mixed-weight modular objects, and perhaps even entirely non-modular objects.