In 1989, Selberg defined what came to be known as the "Selberg class" of $L$-functions, giving rise to a new subfield of analytic number theory in the intervening quarter century. Despite its popularity, a few things have always bugged me about the definition of the Selberg class. I will discuss these nitpicks and describe some modest attempts at overcoming them, with new applications.

# Past Number Theory Seminar

Fix a prime $p$. In this talk, we will discuss the $p$-adic properties of the *coefficients* of the characteristic power series of $U_{p}$ acting on spaces of overconvergent $p$-adic modular forms. These coefficients are, by a theorem of Coleman, power series in the weight variable over $Z_{p}$. Our first goal will be to show that in tame level one, the simplest case, every coefficient is non-zero mod $p$ and then to give some idea of the (finitely many) roots of each coefficient. The second goal will be to explain how it the previous result fails in higher levels, along with possible salvages. This will include revisiting the tame level one case. The progress we've made has applications, and lends understanding, to recent work being made elsewhere on the geometric structure of the eigencurve "near its boundary". This is joint work with Rob Pollack.

Rational points on Kummer varieties can be studied through the variation of Selmer groups of quadratic twists of the underlying abelian variety, using an idea of Swinnerton-Dyer. We consider the case when the Galois action on 2-torsion has a large image. Under a mild additional assumption we prove the Hasse principle assuming the finiteness of relevant Shafarevich-Tate groups. This approach is inspired by the work of Mazur and Rubin.

The classical conjecture of Serre (proved by Khare-Winterberger) states that a continuous, absolutely irreducible, odd representation of the absolute Galois group of Q on two-dimensional F_p-vector space is modular. We show how one can formulate its analogue in characteristic 0. In particular we discuss the weight part of the conjecture. This is a joint work with John Bergdall.

In 1953 Roth proved that any positive density subset of the integers contains a non-trivial three term arithmetic progression. I will present a recent quantitative improvement for this theorem, give an overview of the main ideas of the proof, and discuss its relation to other recent work in the area. I will also discuss some closely related problems.

We discuss a new method to bound the number of primes in certain very thin sets. The sets $S$ under consideration have the property that if $p\in S$ and $q$ is prime with $q|(p-1)$, then $q\in S$. For each prime $p$, only 1 or 2 residue classes modulo $p$ are omitted, and thus the traditional small sieve furnishes only the bound $O(x/\log^2 x)$ (at best) for the counting function of $S$. Using a different strategy, one related to the theory of prime chains and Pratt trees, we prove that either $S$

contains all primes or $\# \{p\in S : p\le x \} = O(x^{1-c})$ for some positive $c$. Such sets arise, for example, in work on Carmichael's conjecture for Euler's function.

Let f be an elliptic modular newform of weight at least 2. The

problem of the automorphy of the symmetric power L-functions of f is a

key example of Langlands' functoriality conjectures. Recently, the

potential automorphy of these L-functions has been established, using

automorphy lifting techniques, and leading to a proof of the Sato-Tate

conjecture. I will discuss a new approach to the automorphy of these

L-functions that shows the existence of Sym^m f for m = 1,...,8.

I'll discuss work (part with Savitt, part with Dembele and Roberts) on two related questions: describing local factors at primes over p in mod p automorphic representations, and describing reductions of local crystalline Galois representations with prescribed Hodge-Tate weights.

In this seminar I will discuss a function field analogue of classical problems in analytic number theory, concerning the auto-correlations of divisor functions, in the limit of a large finite field.