Congruences in modular, Jacobi, Siegel, and mock modular forms with applications

We study congruences in the coefficients of modular and other automorphic forms. Ramanujan famously found congruences for the partition function like p(5n+4) = 0 mod 5. For a wide class of modular forms, we classify the primes for which there can be analogous congruences in the coefficients of the Fourier expansion. We have several applications. We describe the Ramanujan congruences in the counting functions for overparitions, overpartition pairs, crank differences, and Andrews' two-coloured generalized Frobenius partitions. We also study Ramanujan congruences in the Fourier coefficients of certain ratios of Eisenstein series. We also determine the exact number of holomorphic modular forms with Ramanujan congruences when the weight is large enough. In a chapter based on joint work with Olav Richter, we study Ramanujan congruences in the coefficients of Jacobi forms and Siegel modular forms of degree two. Finally, the last chapter contains a completely unrelated result about harmonic weak Maass forms.

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

2010

Author dc:creator

• Dewar, Michael P.

Contributors dc:contributor

• Ahlgren, Scott
• Berndt, Bruce C.
• Dunfield, Nathan M.
• Zaharescu, Alexandru

• Ramanujan congruences
• Tate cycle
• heat cycle
• Fourier coefficients
• Modular forms
• reduced modular forms
• Jacobi forms
• Siegel modular forms

Statement dc:rights

• Copyright Michael P. Dewar

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/16054

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/16054