I will describe joint work with Manjul Bhargava (Princeton) and Tom Fisher (Cambridge) in which we determine the probability that random equation from certain families has a solution either locally (over the reals or the p-adics), everywhere locally, or globally. Three kinds of equation will be considered: quadratics in any number of variables, ternary cubics and hyperelliptic quartics.

# Past Number Theory Seminar

Thanks to the p-adic local Langlands correspondence for GL_2(Q_p), one "knows" all admissible unitary topologically irreducible representations of GL_2(Z_p). In this talk I will focus on some elementary properties of their restriction to GL_2(Z_p): for instance, to what extent does the restriction to GL_2(Z_p) allow one to recover the original representation, when is the restriction of finite length, etc.

In sieve theory, one is concerned with estimating the size of a sifted set, which avoids certain residue classes modulo many primes. For example, the problem of counting primes corresponds to the situation when the residue class 0 is removed for each prime in a suitable range. This talk will be concerned about what happens when a positive proportion of residue classes is removed for each prime, and especially when this proporition is more than a half. In doing so we will come across an algebraic question: given a polynomial f(x) in Z[x], what is the average size of the value set of f reduced modulo primes?

Given an abelian variety A over a number field k, its Kummer variety X is the quotient of A by the automorphism that sends each point P to -P. We study p-adic density and weak approximation on X by relating its rational points to rational points of quadratic twists of A. This leads to many examples of K3 surfaces over Q whose rational points lie dense in the p-adic topology, or in product topologies arising from p-adic topologies. Finally, the same method is used to prove that if the Brauer--Manin obstruction controls the failure of weak approximation on all Kummer varieties, then ranks of quadratic twists of (non-trivial) abelian varieties are unbounded. This last fact arises from joint work with David Holmes.

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.

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.