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

Dewar, Michael P.

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https://hdl.handle.net/2142/16054 Copy

Description

Title

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

Author(s)

Dewar, Michael P.

Issue Date

2010-05-19T18:33:30Z

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

Ahlgren, Scott

Doctoral Committee Chair(s)

Berndt, Bruce C.

Committee Member(s)

• Ahlgren, Scott
• Dunfield, Nathan M.
• Zaharescu, Alexandru

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

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

Abstract

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.

Graduation Semester

2010-5

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

Copyright and License Information

Copyright Michael P. Dewar

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