### Directed Reading and Research - MATH 649T

# Number Theory

## Congruences related to modular forms

### Instructor -- Pavel Guerzhoy

### The class meets on Wednesday, 4 - 5+pm at 413, Keller Hall.

### Wednesday, Oct 26. Guest lecture by Prof. Kaneko, Kyushu University (Japan).

** Office: ** 501, Keller Hall (5-th floor)

** tel: ** (808)-956-6533

** e-mail: ** pavel(at)math(dot)hawaii(dot)edu

Usually, I respond to e-mail messages within a day. If I do not,that means that I do not have a reasonable answer, and am still thinking.

** Office hours: ** Almost every afternoon, weekends included, whenever the participants have questions to ask or discoveries to report.

** General Philosophy **
It is well-known that
the only way to learn an area of mathematics is to try doing your own research in this area.

We do not solve great problems or prove big theorems. That approach, of course, has an advantage: theoretically, one can become a prominent mathematician in just one night. However, this approach also has severe disadvantages. Firstly, it is extremely risky, because a solution of a really serious problem may be a hundred years ahead. Thus, chances are, one will end up with nothing in hand. Secondly, and more importantly, the concentration on a particular question may prevent from learning the whole area of mathematics. However, our primary goal is to learn mathematics, whereas research plays the role of a tool for that.

The participants will work on their own research projects. I will select these projects so that they satisfy the following conditions:

Non-elementary. This means that they require a substantial amount of knowledge on modular forms and related topics. The areas studied in the process of working on a project are important for all participants, and will be the subject the in-class discussions.

Modest and doable. It is much better to succeed at several small projects than to fail at a great one. I will be happy and consider my job done when, several years later, students are ashamed of the papers they published with me.

Decent and non-trivial. The results, if obtained, should be good enough to be publishable in reasonable research journals. This implies taking a certain risk of failure as any kind of research in mathematics involves that. Nobody can say with a certainty how difficult a problem is until the problem is solved. I am not in a position to offer problems whose solutions I know in advance.

** Publish or Perish **
The ultimate goal of a participant is to complete research projects and to publish research papers. This is the only way for a graduate student to become a professional mathematician. There are reasons for that: this is the best way to learn mathematics, and, at the same time, the only way to keep one's chances of continuing a mathematical career in the academy. After a theorem is proved, I will, of course, help to write down a paper, and to submit it to a journal.

** Reading **
This class will be primarily based on the following literature. By no means I suggest to immediately purchase these books. However, I assume that the students have a reasonable access to them via the library, my office, classmates, etc.
Serre, J.-P, * A course in arithmetic, * Translated from French. Graduate Texts in Mathematics, No. 7. Springer-Verlag, New York-Heidelberg, 1973. viii+115 pp.
Koblitz, Neal, * Introduction to elliptic curves and modular forms, * Second edition. Graduate Texts in Mathematics, 97. Springer-Verlag, New York, 1993. x+248 pp. ISBN: 0-387-97966-2
Koblitz, Neal, * p-adic numbers, p-adic analysis, and zeta-functions, * Second edition. Graduate Texts in Mathematics, 58. Springer-Verlag, New York, 1984. xii+150 pp. ISBN: 0-387-96017-1
Silverman, Joseph H., * The arithmetic of elliptic curves, * Corrected reprint of the 1986 original. Graduate Texts in Mathematics, 106. Springer-Verlag, New York, 1992. xii+400 pp. ISBN: 0-387-96203-4
Ono, Ken,
* The web of modularity: arithmetic of the coefficients of modular forms and q-series. *
CBMS Regional Conference Series in Mathematics, 102. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004. viii+216 pp. ISBN: 0-8218-3368-5
Lang, Serge, * Introduction to modular forms.* With appendixes by D. Zagier and Walter Feit. Corrected reprint of the 1976 original. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 222. Springer-Verlag, Berlin, 1995. x+261 pp. ISBN: 3-540-07833-9
Zafier, Don, * Elliptic Modular Forms and Their Applications. * The 1-2-3 of Modular Forms, Springer 2008, p. 1-103.

** Course Objective **
To make it possible for the participants to begin to conduct their own research in the areas
of modular forms, p-adic analysis, and related topics of number theory.

** Prerequisites **
Some knowledge on the graduate level of algebra and complex analysis is required.
Interest in doing mathematics and a desire to work hard are absolutely necessary.

** Format **
The classes will be held in an informal format.
The participants will report on their progress, and put questions. I will make comments trying to explain as much of related material as time allows.
In this way, the discussions become useful for all participants, including myself. Moreover, it is close to real research: a mathematician puts questions and tries to find answers learning in this or that way some mathematics which the researcher (and sometimes anyone else on the planet) knows nothing about beforehand.
Active participants get to develop their own projects (possibly, in collaboration with me).

### Contents

This semester, the intended areas of concentration are related to Weak harmonic Maass forms.
More specifically, we will consider the following:

Congruences discovered by Kaneko, their proofs and generalization

Weight two case: relation elliptic curves, Hodge-Tate decomposition

Zagier' adele