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

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Title: Congruences in modular, Jacobi, Siegel, and mock modular forms with applications
Author(s): Dewar, Michael P.
Director of Research: Ahlgren, Scott
Doctoral Committee Chair(s): Berndt, Bruce C.
Doctoral Committee Member(s): Ahlgren, Scott; Dunfield, Nathan M.; Zaharescu, Alexandru
Department / Program: Mathematics
Discipline: Mathematics
Degree Granting Institution: University of Illinois at Urbana-Champaign
Degree: Ph.D.
Genre: Dissertation
Subject(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.
Issue Date: 2010-05-19
URI: http://hdl.handle.net/2142/16054
Rights Information: Copyright Michael P. Dewar
Date Available in IDEALS: 2010-05-19
Date Deposited: May 2010
 

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