Abstract: The theory of root numbers predicts when elliptic curves over number fields have rational points of infinite order. In this talk, we shall discuss results which bring together the root numbers and non-commutative Iwasawa theory. It is joint work with Coates, Fukaya and Kato, and has connections to some recent work of Rohrlich and T. Dokchitser and V. Dokchitser.

Abstract: Elliptic divisibility sequences are integer recurrence sequences, each of which is associated to an elliptic curve over the rationals together with a rational point on that curve. I'll give the background on these and present a higher-dimensional analogue over arbitrary fields. Suppose E is an elliptic curve over a field K, and P₁, ..., P_n are points on E defined over K. To this information we associate an n-dimensional array of values of K satisfying a complicated nonlinear recurrence relation. These are called elliptic nets. All elliptic nets arise from elliptic curves in this manner. I'll explore some of the properties of elliptic nets and the geometric information they contain, including a connection to generalised Jacobians and the Tate and Weil pairings on the elliptic curve.

Abstract: The Siegel-Walfisz theorem states that for any B>0, we have Σ_{p≤x, p≅d (mod v)} 1 ∼ x/φ(v) log(x) for v ≤ log^B(x) and (v,d)=1. This only gives an asymptotic formula for the number of primes in an arithmetic progression for quite a small modulus v compared to x. However, if we are concerned only with an upper bound, the Brun-Titchmarsh theorem says that for any 1≤v≤x, we have Σ_{p≤x, p≅d (mod v)} 1 << x/φ(v) log(x). In this talk, we will discuss an extension to the Brun-Titichmarsh theorem that concerns the number of integers with exactly s distinct prime factors in an arithmetic progression. This is joint work with Kai Man Tsang and Tsz Ho Chan.

Abstract: The Manin conjectures describe for geometrically easy varieties how the number of their rational points of bounded height should grow as the height bound varies. In this talk I will describe recent computations that suggest a similar statement for K3 surfaces, which are geometrically more complicated. Part of the talk will focus on how to count the number of points in a specific example, using a variation of an algorithm by Noam Elkies.

Abstract: An integer s is called a primitive root modulo a prime p if the multiplicative set generated by s surjects onto all non-zero residue classes modulo p. Artin's primitive root conjecture states that all integers s other than -1 or squares are primitive roots modulo infinitely many primes. I'll discuss a generalization of Artin's primitive root conjecture to number fields and connections this has to the Euclidean Algorithm problem. This is joint work with R. Murty.

Abstract: Random matrix theory has been a hot topic in number theory, particularly since the Rudnick and Sarnak landmark work on the spacing of consecutive zeros of L-functions. This highly accessible talk has a far more elementary flavour, focusing on eigenvalues of random integer matrices instead of the Gaussian Unitary Ensemble. For a fixed n, consider a random n×n integer matrix with entries bounded by the parameter k. I'll give a simple proof that such a matrix almost certainly has no rational eigenvalues (as k increases). Then we'll delve into more detail on the exact eigenvalue distribution of the 2×2 case. Along the way we'll rediscover a forgotten determinant identity and tackle some quadruple sums. This is joint work with Greg Martin.

Abstract: In this talk, we will describe a construction of Eisenstein classes in the parabolic cohomology of Γ₀(M) with values in Milnor's K₂-group of the ring of S-integers of the cyclotomic extension Q(μ_M). In the case where p is an irregular prime, we will explain a congruence between certain projections of Steinberg symbols of p-units and L-values of level-1 cusp forms congruent to an Eisenstein series mod p, a result that was predicted by a conjecture of R. Sharifi.

Abstract: To every hyperbolic 3-manifold M with nonempty boundary, one can associate a pair of related algebraic varieties X(M) and Y(M) called the character varieties of M. These varieties carry much topological information about M, but are in general difficult to compute. In the case that M is a knot complement, X(M) and Y(M) are defined over Q. In this talk, I will discuss how properties of these varieties reflect the topology of M in the case M is a hyperbolic knot complement. I will also show how to obtain explicit equations for the the character varieties associated to a bi-infinite family of hyperbolic knots K(m,n) and discuss consequences of these results related to the existence of integral points on these curves. This is joint work with Kate Petersen and Ronald van Luijk.

Abstract: For a fixed prime p, we consider the space S(N,k) of cusp forms of weight k and level N, with N coprime to p. In 1995, J.-P. Serre proved the existence of a measure u_p with respect to which the eigenvalues of the pth Hecke operator acting on S(N,k) are equidistributed as k+N tends to infinity. We will derive an effective version of Serre's theorem and apply it to study the factorization of J₀(N) into simple abelian varieties. Our methods can also be applied to study the variation of eigenvalues of the Frobenius automorphism acting on a family of curves mod p and the variation of eigenvalues of adjacency matrices of regular graphs. (This is joint work with Kaneenika Sinha.)

Abstract: In this expository talk, I'll be discussing the work of Granville and Soundararajan on sums of multiplicative functions in arithmetic progressions.

Abstract: Assuming the Riemann Hypothesis, we establish lower bounds for moments of the derivative of the Riemann zeta-function averaged over the non-trivial zeros of zeta. Our proof is based upon a method of Rudnick and Soundararajan that provides analogous bounds for moments of L-functions at the central point. This is joint work with Milinovich.

Abstract: A classical theorem due to Linnik gives a bound for the least prime number in an arithmetic progression. Lagarias, Montgomery and Odlyzko gave a generalization of this result to any number field. Their proof relies on some results on the distribution of the zeros of the Dedekind zeta function (zero-free regions, Deuring-Heilbronn phenomenon). In this talk, I will present some new results about these zeros. As a consequence, we are able to prove an effective version of the theorem of Lagarias et al.