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Title:Problems in number theory and hyperbolic geometry
Author(s):Sinick, Jonah D.
Director of Research:Dunfield, Nathan M.
Doctoral Committee Chair(s):Leininger, Christopher J.
Doctoral Committee Member(s):Dunfield, Nathan M.; Ahlgren, Scott; Alexander, Stephanie B.
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Subject(s):Number theory
Modular forms
Hyperbolic geometry
Low dimensional topology
Abstract:In the first part of this thesis we generalize a theorem of Kiming and Olsson concerning the existence of Ramanujan-type congruences for a class of eta quotients. Specifically, we consider a class of generating functions analogous to the generating function of the partition function and establish a bound on the primes ℓ for which their coefficients c(n) obey congruences of the form c(ℓn + a) ≡ 0 (mod ℓ). We use this last result to answer a question of H.C. Chan. In the second part of this thesis [S2] we explore a natural analog of D. Calegari’s result that there are no hyperbolic once-punctured torus bundles over S^1 with trace field having a real place. We prove a contrasting theorem showing the existence of several infinite families of pairs (−χ, p) such that there exist hyperbolic surface bundles over S^1 with trace field of having a real place and with fiber having p punctures and Euler characteristic χ. This supports our conjecture that with finitely many known exceptions there exist such examples for each pair ( −χ, p).
Issue Date:2011-08-25
Rights Information:Copyright 2011 Jonah Sinick
Date Available in IDEALS:2011-08-25
Date Deposited:2011-08

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