Login | Register

Overconvergent Modular Forms, Theoretical and Computational Aspects


Overconvergent Modular Forms, Theoretical and Computational Aspects

Johnson, Alexandre (2022) Overconvergent Modular Forms, Theoretical and Computational Aspects. Masters thesis, Concordia University.

[thumbnail of Johnson_MSc_F2022.pdf]
Text (application/pdf)
Johnson_MSc_F2022.pdf - Accepted Version
Available under License Spectrum Terms of Access.


In this thesis, we perform a review of the theory of overconvergent modular forms, then we explore the distribution of the eigenvalues of the Hecke operator T_p by considering their p-adic valuations. We begin by covering algebraic and geometric definitions of modular forms, then expanding these definitions to overconvergent modular forms. We then introduce algorithms, from "Computations with classical and p-adic modular forms" by Alan G. B. Lauder, which provide a method for calculating the p-adic valuations of the aforementioned eigenvalues. In order to implement these algorithms, programs were written for the Sagemath computer algebra program to perform the necessary calculations. These programs were used to collect lists of p-adic valuations, for various values of p and for spaces of modular forms of various weights and of various levels. The collected data confirms the fact that the Gouvea-Mazur conjecture is false, but also indicates that it may be a useful approximation of the true behavior at large weights or at large values of p, at least for the first few slopes. It shows the existence of "plateaus" of weights which have the same slopes, up to the precision used, even at low values of p and k. The reason for the existence of these "plateaus" is unknown.

Divisions:Concordia University > Faculty of Arts and Science > Mathematics and Statistics
Item Type:Thesis (Masters)
Authors:Johnson, Alexandre
Institution:Concordia University
Degree Name:M. Sc.
Date:August 2022
Thesis Supervisor(s):Iovita, Adrian and Rosso, Giovanni
ID Code:990948
Deposited By: Alexandre Johnson
Deposited On:27 Oct 2022 14:28
Last Modified:27 Oct 2022 14:28


[1] Jose Ignacio Burgos Gil and Ariel Pacetti. Hecke and Sturm bounds for Hilbert modular forms over real quadratic fields. Math. Comp., 86(306):1949–1978, 2017.
[2] Fred Diamond and Jerry Shurman. A first course in modular forms, volume 228 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2005.
[3] Fernando Q. Gouvˆea. p-adic numbers. Universitext. Springer-Verlag, Berlin, second edition, 1997. An introduction.
[4] Robin Hartshorne. Algebraic geometry. Graduate Texts in Mathematics, No. 52. Springer- Verlag, New York-Heidelberg, 1977.
[5] Nicholas M. Katz. p-adic properties of modular schemes and modular forms. In Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture Notes in Math., Vol. 350, pages 69–190. Springer, Berlin, 1973.
[6] Alan G. B. Lauder. Computations with classical and p-adic modular forms. LMS J. Comput. Math., 14:214–231, 2011.
[7] Yu. I. Manin. Selected papers of Yu. I. Manin, volume 3 of World Scientific Series in 20th Century Mathematics. World Scientific Publishing Co., Inc., River Edge, NJ, 1996.
[8] Jean-Pierre Serre. Endomorphismes compl`etement continus des espaces de Banach p-adiques. Inst. Hautes ́Etudes Sci. Publ. Math., (12):69–85, 1962.
[9] Joseph H. Silverman. The arithmetic of elliptic curves, volume 106 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1992. Corrected reprint of the 1986 original.
[10] The Stacks project authors. The stacks project. https://stacks.math.columbia.edu/tag/ 073K, 2022.
[11] William Stein. Modular forms, a computational approach, volume 79 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2007. With an appendix by Paul E. Gunnells.
[12] Daqing Wan. Dimension variation of classical and p-adic modular forms. Invent. Math., 133(2):449–463, 1998
All items in Spectrum are protected by copyright, with all rights reserved. The use of items is governed by Spectrum's terms of access.

Repository Staff Only: item control page

Downloads per month over past year

Research related to the current document (at the CORE website)
- Research related to the current document (at the CORE website)
Back to top Back to top