A Guide to Arithmetic Geometry

Many of the recent posts on this blog have been fairly advanced and specialized, but I still want this blog to be useful to beginners, especially to beginners in arithmetic geometry (which is the subfield I am specializing in). So I decided to compile some of my earlier arithmetic geometry posts on this page in an order which I think would be good to follow for a beginner. My hope is that it serves as a kind of roadmap to arithmetic geometry. It is important to note that this is not the only path to follow in arithmetic geometry; but this is perhaps the path that I have personally followed, reflected in this blog.

Note that, some of these posts being older, there is no guarantee on the quality of the content. I always suggest consulting the more authoritative references linked at the end of the posts. If there are any glaring errors, please let me know! With that out of the way, here is my guide to arithmetic geometry!

I. Elliptic Curves

I believe one of the best possible introductions to arithmetic geometry is via the theory of elliptic curves. It is one of the simpler objects one encounters, but has many fascinating properties we may want to study in greater abstraction. The best part is that one does not need a thorough knowledge of algebraic geometry or number theory to start learning about elliptic curves (though they are needed if one wants to go deeper). Here is my first post on elliptic curves:

I.1 Elliptic Curves

There are two very important aspects of the theory of elliptic curves that I want to highlight (there are others, but in this blog I mostly focus on these two). The first aspect is that one can obtain a Galois representation known as the Tate module via the torsion points of elliptic curves. I discuss this (together with other ways of obtaining Galois representations) in this post:

I.2 Galois Representations

The other aspect of the theory of elliptic curves that I want to highlight is the fact that elliptic curves over the complex numbers are complex tori, and that one can use this fact to describe the moduli space of all elliptic curves over the complex numbers. It is discussed in this post:

I.3 The Moduli Space of Elliptic Curves

II. Modular Forms

Once one knows the basics of the theory of elliptic curves, a good follow-up is the theory of modular forms. All that one needs to know the definition of a modular form is some basic complex analysis and a little group theory (namely of \mathrm{SL}_{2}(\mathbb{R}) and \mathrm{SL}_{2}(\mathbb{Z})). They are holomorphic functions on the upper-half plane that transform in a specific way under the action of \mathrm{SL}_{2}(\mathbb{Z}) or one of its congruence subgroups and are bounded as the imaginary part of its argument goes to infinity. However, it is not immediately obvious from this definition how modular forms are related to elliptic curves. To see this, one needs to use the theory of the moduli space of elliptic curves mentioned earlier (possibly with some “level structure”), and see modular forms as sheaves on modular curves, which are compactifications of these moduli spaces that can be expressed as an algebraic variety (a shape defined by polynomials). Here is my post on modular forms:

II.1 Modular Forms

A very important aspect of modular forms is that the vector spaces they form have certain special endomorphisms known as Hecke operators. There are many ways to describe these operators, and they have many fascinating properties, and some (but not all) of them are described in this post:

II.2 Hecke Operators

Just as one can obtain Galois representations from elliptic curves, one can also obtain Galois representations from modular forms! The famous Shimura-Taniyama-Weil conjecture (now a theorem) that was used to prove Fermat’s Last Theorem states that Galois representations coming from elliptic curves (over the rational numbers) also come from certain modular forms (namely eigenforms of weight 2). One way to obtain Galois representations from these particular modular forms (this method is also known as the Eichler-Shimura construction) is described in this post:

II.3 Galois Representations Coming From Weight 2 Eigenforms

Other than Galois representations, another number theoretic object closely connected to modular forms is L-functions. Here is a pair of introductory posts (they are pretty independent of the last two paragraphs, i.e. the posts labeled II.2 and II.3) I wrote that describes these connections:

II.4 Bernoulli Numbers, Fermat’s Last Theorem, and the Riemann Zeta Function

II.5 Iwasawa theory, p-adic L-functions, and p-adic modular forms

III. Automorphic Forms

Modular forms are but a special case of a more general idea that comes from representation theory, that of automorphic forms. Here is an introductory post I wrote that leads up to the ideas involved:

III.1 Representation Theory and Fourier Analysis

The next post I will link here defines automorphic forms properly. An important part of this next post is showing how a modular form is an automorphic form. Note that automorphic forms are usually defined using very different language compared to how modular forms are usually defined (because automorphic forms come from representation theory!) so it is quite important to be able to make the connection. Here is my post on automorphic forms:

III.2 Automorphic Forms

A running theme of this “guide” so far has been Galois representations. And just as modular forms can give rise to Galois representation, the same is also expected of automorphic forms. Furthermore, the opposite direction is also expected to hold in some form. This, very very roughly, is one form of the Langlands correspondence. So far I have not written a post on the global Langlands correspondence in the number field case (which generalizes the Shimura-Taniyama-Weil conjecture), but I have written the following posts on the local field case and the global function field (over a finite field) case:

III.3 The Local Langlands Correspondence for General Linear Groups

III.4 The Global Langlands Correspondence for Function Fields over a Finite Field

In lieu of a post on the global Langlands correspondence for the number field case, since I have not written one yet, I will link here instead to the slides of an introductory talk on the Langlands program (which does include a discussion of the global Langlands correspondence for the number field case) I gave at the What is… a Seminar? series:

III.5 What is… the Langlands Program (Talk Slides)

Other guides to arithmetic geometry

All of what I’ve described here is just one of many, many ways to explore the vast landscape of arithmetic geometry. There are many other aspects that have not been discussed on this blog. I am furthermore limited to the blog posts that I have already written (I might update this page when I write more posts that fit into the idea of this page). I will post here links to posts or pages of a similar nature to what I’ve written here from other arithmetic geometers:

Matthew Emerton’s Comment on Terence Tao’s Blog Post “Learn and Relearn Your Field”

David Zureick-Brown’s Advice to Potential Students in Arithmetic Geometry