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Coefficients Of Modular Forms And Applications To Partition Theory, Michael Anthony Hanson
Coefficients Of Modular Forms And Applications To Partition Theory, Michael Anthony Hanson
Doctoral Dissertations
We begin with an overview of the theory of modular forms as well as some relevant sub-topics in order to discuss three results: the first result concerns positivity of self-conjugate t-core partitions under the assumption of the Generalized Riemann Hypothesis; the second result bounds certain types of congruences called "Ramanujan congruences" for an infinite class of eta-quotients - this has an immediate application to a certain restricted partition function whose congruences have been studied in the past; the third result strengthens a previous result that relates weakly holomorphic modular forms to newforms via p-adic limits.
On The Iwasawa Invariants Of Nonordinary Modular Forms, Rylan J. Gajek-Leonard
On The Iwasawa Invariants Of Nonordinary Modular Forms, Rylan J. Gajek-Leonard
Doctoral Dissertations
We extend known results on the behavior of Iwasawa invariants attached to Mazur-Tate elements for p-nonordinary modular forms of weight k=2 to higher weight modular forms with a_p=0. This is done by using a decomposition of the p-adic L-function due to R. Pollack in order to construct explicit lifts of Mazur-Tate elements to the full Iwasawa algebra. We then study the behavior of Iwasawa invariants upon projection to finite layers, allowing us to express the invariants of Mazur-Tate elements in terms of those coming from plus/minus p-adic L-functions. Our results combine with work of Pollack and Weston to relate the …