Modular forms, the Shimura correspondence, and arithmetic applications

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 Q³n+β₀) \equiv c \cdot p(\ell Q n+β₁) 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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2023

Author dc:creator

• Dicks, Robert

Contributors dc:contributor

• Ahlgren, Scott
• Ford, Kevin
• Zaharescu, Alexandru
• Thorner, Jesse

• Modular Forms
• Partitions

Statement dc:rights

• Copyright 2023 Robert Dicks

Language dc:language

en, eng

Handle dc:identifier

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

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F120376