Modular forms, the Shimura correspondence, and arithmetic applications

Dicks, Robert

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https://hdl.handle.net/2142/120376 Copy

Description

Title

Modular forms, the Shimura correspondence, and arithmetic applications

Author(s)

Dicks, Robert

Issue Date

2023-04-23

Director of Research (if dissertation) or Advisor (if thesis)

Ahlgren, Scott

Doctoral Committee Chair(s)

Ford, Kevin

Committee Member(s)

• Zaharescu, Alexandru
• Thorner, Jesse

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Modular Forms
• partitions

Abstract

In this thesis, we prove results on modular forms with special emphasis on their arithmetic properties. In the second chapter, we bound the order of vanishing at infinity for certain spaces of cusp forms of even weight $k \geq 4$. This generalizes a theorem of Ogg on whether or not $\infty$ is a Weierstrass point on certain modular curves. In the third chapter, we classify congruences for a wide range of spaces of half-integral weight forms on $\operatorname{SL}_2(\mathbb{Z})$ which are supported on finitely many square classes modulo a prime $\ell \geq 5$; the main result can be viewed as a modulo $\ell$ analogue of a similar result of Vign\`{e}ras in characteristic $0$. In the fourth chapter, we express weight $2$ CM newforms which are eta quotients as $p$-adic limits of the derivatives of the Weierstrass mock modular forms associated to their elliptic curves. In the fifth chapter, for a prime $\ell \geq 5$ and a wide range of $c \in \mathbb{F}_\ell$, we prove congruences of the form $p(\ell Q^3n+\beta_0) \equiv c \cdot p(\ell Q n+\beta_1)$ for infinitely many primes $Q$. Here, $p(n)$ denotes the partition function. For $r \in \mathbb{Z}^+$, we prove similar congruences for the $r$-colored partition function $p_r(n)$. . The chapters of this thesis are self-contained; each chapter is based on a different paper. In particular, the notation will vary from chapter to chapter.

Graduation Semester

2023-05

Type of Resource

Thesis

Handle URL

https://hdl.handle.net/2142/120376

Copyright and License Information

Copyright 2023 Robert Dicks

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