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Automorphic L-Functions, Geometric Invariants, and Dynamics


Perozim de Faveri, Alexandre (2022) Automorphic L-Functions, Geometric Invariants, and Dynamics. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/e1zd-7v46.


We address three different problems in analytic number theory.

In the first part, we show that the completed L-function of a modular form has Ω(Tδ) simple zeros with imaginary part in [-T, T], for any δ < 227. This is the first power bound for forms with non-trivial level in this problem, where previously the best result was Ω(log log log T). Along the way, we also improve the corresponding bound in the case of trivial level, and sharpen a certain zero-density result.

In the second part, we study the variance for the distribution of closed geodesics in random balls on the modular surface. A probabilistic model in which closed geodesics are modeled using random geodesic segments is proposed, and we rigorously analyze this model using mixing of the geodesic flow. This leads to a conjecture for the asymptotic behavior of the variance, and we prove this conjecture for sufficiently small balls.

In the third part, we prove Sarnak's Möbius disjointness conjecture for C1+ε skew products on the 2-torus over a rotation of the circle.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:L-functions; geometric invariants; Möbius disjointness
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Awards:Scott Russell Johnson Graduate Dissertation Prize in Mathematics, 2022. Scott Russell Johnson Prize for Excellence as a First-Year Graduate Student, 2019.
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Radziwiłł, Maksym
Thesis Committee:
  • Ramakrishnan, Dinakar (chair)
  • Dunn, Alexander
  • Michel, Philippe R.
  • Radziwiłł, Maksym
Defense Date:16 May 2022
Record Number:CaltechTHESIS:05182022-065221100
Persistent URL:
Related URLs:
URLURL TypeDescription adapted for Chapter 2 and Appendix A adapted for Chapter 3 adapted for Chapter 4
Perozim de Faveri, Alexandre0000-0001-7180-9382
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:14585
Deposited By: Alexandre Perozim de Faveri
Deposited On:23 Jun 2022 20:45
Last Modified:03 Aug 2022 23:25

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