Research Interests

My area of research is number theory, especially the theory of automorphic forms. I am also
interested in several related areas, such as algebraic geometry and representation theory.

``It is a deeper subject than I appreciated and, I begin to suspect, deeper than anyone yet appreciates.
To see it whole is certainly a daunting, for the moment even impossible, task.''

(Robert Langlands, writing about the theory of automorphic forms.)

Preprints and Papers

They are available here.

For students

I have made several posts on Mathoverflow and MathSE related to my areas of research interest.
Here are links to some that may be helpful to students, especially those interested in working with me.

Posts on algebraic geometry:

Some thoughts on learning algebraic geometry, including the merits of studying EGA as a grad student
and the relationship between commutative algebra and algebraic geometry.

The importance of the Hodge conjecture.

An intuitive explanation of divisors on and genus of a curve.

Another discussion of the genus of plane curves, this time in the context of integrating algebraic functions.

A discussion of stable curves and semistable rduction.

An explanation of the group law on elliptic curves from a historical perspective, and a discussion of how it generalizes to the context of higher genus curves.

A discussion of elliptic integrals from a geometric perspective.

A concrete proof that an elliptic curve injects into its Picard group.

An explanation of Cartier divisors and the relationship to line bundles.

How to think about and work with the Zariski topology. (And some related remarks.)

Some discussions of the Nullstellensatz: one and two.

A proof of Noether normalization via projective geometry.

Some remarks on the role of sheaves in geometry.

A short proof that an irreducible projective algebraic group has to be abelian.

Two proofs that morphisms of elliptic curves necessarily preserve the group structure. (The first is closely related to the argument of the preceding post.)

Some remarks on how to think about smooth and flat morphisms in algebraic geometry.

A concrete example of faithfully flat descent.

A brief discussion of semi-stable families.

Some thoughts on non-commutative algebraic geometry.

Posts on number theory:

A brief description of some of the topics studied by number theorists, and their interrelations.

A possible reading list for learning about Galois representations.

The significance of Tate's thesis.

The meaning of Tate twists in etale cohomology, and of the Tate conjecture.

An example (related to Mazur's celebrated theorem describing the possible rational torsion points
on elliptic curves over the rationals) of how Grothendieckian algebraic geometry can help in solving Diophantine equations.

A computation of the 2-torsion group scheme of an elliptic curve over Z[1/3].

An explanation of the geometric point of view on modular forms.

A discussion of the result of Abrashkin and Fontaine on the non-existence of abelian schemes over Spec Z.

Some introductory discussions on the Langlands program and its relationship to number theory: a discussion of reciprocity, especially for n = 1 and 2,
and another, and some general remarks on functoriality and reciprocity. (And another remark related to reciprocity for two-dimensional Artin representations.)

Some more posts discussing the Langlands program and its relationship to number theory: a concrete example of reciprocity for GL(2),
more on reciprocity for GL(2), some comments on reciprocity for GL(n) with n bigger than 2, and some brief examples of concrete consequences of functoriality and reciprocity.

Here is a post related to the first of the preceding four posts, discussing some explicit instances of abelian and non-abelian class field theory.

And here is a concrete example of Jacquet--Langlands functoriality.

Here is an analysis and summary of a paper of Taylor and Yoshida (and a related set of notes by Yoshida) proving local-global compatibility for GL(n). (It is rather technical.)

Some technical remarks about the difficulty of proving reciprocity between automorphic forms and Galois representations over non-totally real fields.

Here are some slightly technical remarks on Weil groups: one and two; and a brief explanation of the hypothetical Langlands group.

A somewhat technical discussion of the local Langlands correspondence at archimedean primes.

Here is a historical tour through the theory of automorphic forms. (It is likely inaccurate in many respects; it conveys my sense of the subject,
such as it is, informed by my own reading as well as what I've learned from others.) Here is something similar focused on modular forms.

Some remarks on Hecke operators and Hecke correspondences.

A comment on the Eichler basis problem and its relationship to Jacquet--Langlands transfer.

A special case of the Siegel--Weil formula, described somewhat geometrically.

Here is brief discussion on the properties of L-functions. Here is another, and another.

Here is a brief overview of the significance of p-adic L-functions in number theory.

A discussion of how to apply the Kronecker--Weber theorem in practice, including illustrations of how to use the conductor-discriminant formula.
And an even more concrete discussion in the case of quadratic fields.

An explicit example of class field theory.

A historical review of the ideas underlying the modern theory of modular curves and (more generally) Shimura varieties. And a short discussion
of the motivations for studying them.

The basic idea behind the Eichler--Shimura relation for the Hecke operator at p.

A brief description of the BSD conjecture for elliptic curves.

A concise description of Tate's geometric reformation of the BSD conjecture in the function field case, together with the main conjecture of Iwasawa theory for elliptic curves,
which is an analogue of Tate's conjecture in the context of elliptic curves over number fields.

Some remarks on the Mordell conjecture, Chabauty's method, and Minhyong Kim's anabelian generalization thereof.

A brief discussion of Dwork's p-adic methods in arithmetic geometry, and the modern theories that they helped to give rise to. (My contribution to this discussion gives a related
overview of the historical context of the development of p-adic Hoge theory.)

A discussion of the elementary proof of the prime number theorem and its historical and mathematical significance.

My feelings about morally correct proofs in number theory.

Posts on representation theory:

Some remarks on the relationships between representation theory and harmonic analysis (and some more such remarks here ),
and some related comments on the importance of representation theory.

A brief description of the problem of classifying the unitary dual of Lie groups.

An introduction to locally symmetric spaces (often refered to as ``G mod K mod Gamma''s).