Problems in number theory and hyperbolic geometry

Sinick, Jonah D.

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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

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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