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

William Craig

TL;DR

This work introduces a novel $p$-adic transfer mechanism to derive Sturm-type bounds by lifting known bounds from one graded space to another using purely arithmetic input. The author applies this to the space of level-one quasimodular forms, obtaining uniform bounds for coefficients in ${ obreakb Z}$ and ${ obreakb Z/mb Z}$ and proving two main results: a polynomial-size bound for the order of vanishing (Theorem T: Main) and a bound valid modulo primes (Theorem T: Sturm Q Only). The method blends explicit coefficient-bounds, linear-algebra rank arguments modulo many primes, and a recursive descent that yields an equilibrium bound, with potential applicability to other quasi- and mixed-weight objects. The results pave the way for uniform computational treatments of quasimodular forms and related $q$-series, highlighting both robustness and future extension opportunities in broader modular-context settings.

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.

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

TL;DR

This work introduces a novel -adic transfer mechanism to derive Sturm-type bounds by lifting known bounds from one graded space to another using purely arithmetic input. The author applies this to the space of level-one quasimodular forms, obtaining uniform bounds for coefficients in and and proving two main results: a polynomial-size bound for the order of vanishing (Theorem T: Main) and a bound valid modulo primes (Theorem T: Sturm Q Only). The method blends explicit coefficient-bounds, linear-algebra rank arguments modulo many primes, and a recursive descent that yields an equilibrium bound, with potential applicability to other quasi- and mixed-weight objects. The results pave the way for uniform computational treatments of quasimodular forms and related -series, highlighting both robustness and future extension opportunities in broader modular-context settings.

Abstract

Sturm's theorem states that a modular form with coefficients in or 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 ``-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 or , which extends the non-uniform results for 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.
Paper Structure (25 sections, 16 theorems, 110 equations)

This paper contains 25 sections, 16 theorems, 110 equations.

Table of Contents

  1. Introduction and Statement of results
  2. Nuts and Bolts
  3. Modular forms basics
  4. Modular forms modulo $p$
  5. Some facts on primes
  6. Bounds on coefficients of quasimodular forms
  7. Some linear algebra
  8. Proof of Theorem \ref{['T: Main']}
  9. Stage 1: From mere existence to an explicit bound
  10. Stage 2: Recursive descent
  11. Stage 3: Final steps of reducing bound
  12. Proof of Theorem \ref{['T: Sturm Q Only']}
  13. Stage 1: From mere existence to an explicit bound
  14. Stage 2: Recursive descent
  15. Stage 3: Final steps of reducing bound
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $k \geq 1$ be an integer and let $\Gamma$ be a congruence subgroup of $\mathrm{SL}_2({\mathbb Z})$ with finite index $[\mathrm{SL}_2({\mathbb Z}):\Gamma]$. Let $f$ be a holomorphic modular form for $\Gamma$ of weight $k$ with integer coefficients. The following are true:

Theorems & Definitions (33)

  • Theorem 1.1: OnoWeb
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5: OnoWeb
  • Lemma 2.6
  • ...and 23 more