# A second course in modular forms

I was wondering what would be a suitable project/ reading to do as a second outing in Modular Forms. I have read Serre's book "A course in arithmetic" in a Topics course last semester, right upto (and including) Hecke operators.

Suggestions I have received till now are mostly geared towards reading Diamond-Shruman "A first course in modular forms". But as I understand, this is a serious graduate level textbook and would be too dense for my final semester Undergraduate project.

Left to myself, I would happily explore more of congruence subgroups and higher level modular forms from Koblitz' book "Introduction to elliptic curves and modular forms" and supplement it with problems from Murty's "Problems in the theory of Modular forms". Does this sound like a good plan to you ? Please help with more suggestions, especially in a project flavor (even a small side project would do).

EDIT: Knowing my background in number theory and related areas might help:

Algebra- Abstract algebra, Field and Galois theory, Representations of finite groups, Commutative algebra

Algebraic number theory- Number fields, number rings, Ideal Class group, Dirichlet's Unit theorem, Quadratic reciprocity, p-adic fields and rings, Hensels lemma, quadratic forms

Analytic number theory- Reimann-Zeta function, L-functions, Dirichlet's theorem on AP's, Modular forms

Analysis- Complex Analysis, Fourier analysis on the circle and finite Fourier, Measure theory

• If this is for an undergraduate project then surely you have an advisor? Why not ask them? – Will R Jan 23 '17 at 10:22
• Actually, MSE offers several topics. Did you have already a look? For example, this project. – Dietrich Burde Jan 23 '17 at 10:45
• Try The 1-2-3 of modular forms by Don Zagier. – Jack D'Aurizio Jan 23 '17 at 10:51
• @WillR This is indeed for an undergraduate project. Talking to my advisor is where the suggestions for Diamond-Shruman came from. He also agrees that reading Koblitz' book should predate this. What I am looking for is more in the flavor of how modular forms merges with other parts of number theory- the Jacobi Four Squares theorem and the more generalized problem of number of representations, for example – Manas Jan 23 '17 at 13:17
• @JackD'Aurizio I have begun this book last week. Its a delightful read and brings together a lot of what I was looking for. Thanks – Manas Jan 23 '17 at 13:20

A lot depends on what most interests you about modular forms. Do you care more about the analytic aspects or algebraic aspects? Do you want to learn about modular forms for their own sake, or are you interested in where they fit in the wider world of number theory and their applications to, for example, Fermat's last theorem?

I haven't read Koblitz, but my (possibly flawed) understanding is that it is slightly more analytic, for example looking at things like half weight modular forms. It is quite a gentle introduction. Zagier's book is similar in this respect.

Based on your background, you stand a good chance with Diamond and Shurman. It is not actually a book about modular forms; rather it is an introduction to the modularity theorem (formerly the Taniyama-Shimura conjecture). As such, there is plenty you can skip and still come away with a detailed knowledge of modular forms as well as a good grasp of the ideas that go into the proof of Fermat's last theorem.

Chapters 1 and 2 are the introduction to modular forms over congruence subgroups that you're looking for.

Chapter 3 gives dimension formulas for spaces of modular forms and can happily be skimmed. I would skip chapter 4 and study chapter 5 which is about Hecke operators. The key point is that the modular forms which are eigenvalues for all the Hecke operators form an orthogonal basis for the space of modular forms.

Chapters 6-8 are tough, but if you're willing to take on trust that we can attach Galois representations to modular eigenforms, then you can readily skip to chapter 9 (you'll have to skip some of the proofs in $\S$9.5), which is a gentle introduction to Galois representations and their relation to Fermat's last theorem and the modularity theorem.

• As you can probably gauge, I am more of an algebraic person. But having recently read some analytic number theory, I now feel that its not wise to ignore the analytic view completely. That said, in modular forms I am more interested in where they fit into the wider world of number theory. From what you say, Diamond-Shruman is what I have been looking for. Chapters 1-3 seem like a good broad plan for the semester. Thanks for the roadmap! – Manas Jan 25 '17 at 18:55
• @ManasNethil Perheps even skip chapter 3 in favour of chapter 5. It's the Hecke theory that makes modular forms of interest to algebraic number theorists. There are also parts of chapter 1 and 2 which can be skipped or skimmed ($\S$ 1.5 for instance). At least for me, the relationship to Galois representations and Fermat's last theorem is the most interesting aspect of modular forms. If you get a chance to skim chapter 9, (particularly 9.3, 9.4 and 9.6) it's well worth it. – Mathmo123 Jan 25 '17 at 19:06