Modular forms, the Shimura correspondence, and arithmetic applications
Dicks, Robert
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https://hdl.handle.net/2142/120376
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 $p_r(n)$.
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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.
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