### Directed Reading and Research - MATH 699

# Number Theory

## \(p\)-adic modular forms

### Instructor -- Pavel Guerzhoy

### The class meets whenever the participants have questions to ask or discoveries to report

** 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 **
This is a reading class with the objective to learn \(p\)-adic theories of modular forms.

It is supposed that students study the literature listed below during the semester. The reading should be smooth: questions should be immediately
addressed to me, and discussed so that the student never gets stuck, and continues to study. The literature is organized and should be studied in
a certain order.

** Literature **
This list will be updated in the course of semester.

** Preparatory part **
These texts are assumed to be well-known, and should only be consulted as handbooks. They contain all necessary definitions and principal theorems concerning the subjects involved.

** Modular forms **
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. ** Chapter VII ** only.
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
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
Zagier, Don, * Elliptic Modular Forms and Their Applications. * The 1-2-3 of Modular Forms, Springer 2008, p. 1-103.

** \(p\)-adic stuff **
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
Courtieu, Michel; Panchishkin, Alexei * Non-Archimedean L-functions and arithmetical Siegel modular forms, * Second edition. Lecture Notes in Mathematics, 1471. Springer-Verlag, Berlin, 2004. viii+196 pp. ISBN: 3-540-40729-4 ** Chapter I ** only

** All in one, and more **

Lecture notes by Frank Calegari from an Arizona school are
available on the web.
These notes are devoted to the same target as our class. Assume, however, a more modest target: we want
to become able to begin reading Calegari's text as a result of this class (and, probably, a class in algebraic geometry).

** Serre's theory **
should be learned from the original paper. This is easy to read piece of classics.
Serre, Jean-Pierre; * Formes modulaires et fonctions zêta p-adiques, * Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972), pp. 191–268. Lecture Notes in Math., Vol. 350, Springer, Berlin, 1973.

** Katz' theory **
is a bit more difficult since it requires some knowledge of mathematics. However, at least the statements of the theorems are
accessible.
Katz, Nicholas M.; * p-adic interpolation of real analytic Eisenstein series.* Ann. of Math. (2) 104 (1976), no. 3, 459–571.

** \(p\)-adic families of modular forms **
Despite of the word "elementary" in the title, already the simplest approach to Hida's Control Theorem
will seem to you neither elementary nor easy. However, the theorem itself is striking. That is Chapter 7 from
Hida, Haruzo;
* Elementary theory of L-functions and Eisenstein series.*
London Mathematical Society Student Texts, 26. Cambridge University Press, Cambridge, 1993. xii+386 pp.
It is now natural to ask about what happens if the condition of being \(p\)-ordinary is relaxed. You will find a brief account of that
(along with many things which you have already learned so far) in
Gouvêa, Fernando Q.;* Arithmetic of p-adic modular forms.* Lecture Notes in Mathematics, 1304. Springer-Verlag, Berlin, 1988. viii+121 pp. ISBN: 3-540-18946-7