University of Illinois Urbana-Champaign

Problems in number theory and hyperbolic geometry

Sinick, Jonah D.

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Sinick_Jonah.pdf

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https://hdl.handle.net/2142/26182

Description

Title
Problems in number theory and hyperbolic geometry
Author(s)
Sinick, Jonah D.
Issue Date
2011-08-25T22:17:39Z
Director of Research (if dissertation) or Advisor (if thesis)
Dunfield, Nathan M.
Doctoral Committee Chair(s)
Leininger, Christopher J.
Committee Member(s)
  • Dunfield, Nathan M.
  • Ahlgren, Scott
  • Alexander, Stephanie B.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Date of Ingest
2011-08-25T22:17:39Z
Keyword(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).
Graduation Semester
2011-08
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http://hdl.handle.net/2142/26182
Copyright and License Information
Copyright 2011 Jonah Sinick

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