2026-09-19T07:08:12Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/120376 2023-09-05 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

Ahlgren, Scott Ford, Kevin Zaharescu, Alexandru Thorner, Jesse 2023-05 application/pdf en text Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2025-05-01 The student, Robert Dicks, accepted the attached license on 2023-04-21 at 04:54. The student, Robert Dicks, submitted this Dissertation for approval on 2023-04-21 at 04:59. This Dissertation was approved for publication on 2023-04-23 at 12:00. DSpace SAF Submission Ingestion Package generated from Vireo submission #19003 on 2023-09-01 at 17:13:33 Modular forms, the Shimura correspondence, and arithmetic applications Dicks, Robert 2023-04-23 Modular Forms Partitions 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. Thesis eng https://hdl.handle.net/2142/120376 Copyright 2023 Robert Dicks Ph.D. Dissertation Mathematics University of Illinois at Urbana-Champaign Mathematics