Number Theory

The study of Egyptian fractions, representing rational numbers as the sum of 1/n for distinct denominators, is one of the oldest areas of number theory. In this talk we will survey some of open problems in the area, from the infamous Erdős-Straus conjecture to some which are lesser known, and discuss some recent progress on other problems, including the solution to the Erdős-Graham conjecture, that 1 can be written as the sum of 1/n where n is drawn from an arbitrary set of integers of positive density.

A set of integers greater than 1 is primitive if no member in the set divides another. Erdős proved in the 1930s that the sum of 1/(a log a), ranging over a in A, is uniformly bounded over all choices of primitive sets A. In the 1980s he asked if this bound is attained for the set of prime numbers. In this talk we describe recent work which answers Erdős’ conjecture in the affirmative. We will also discuss applications to old questions of Erdős, Sárközy, and Szemerédi from the 1960s.

Given a bounded submanifold M in R^n, how many rational points with common bounded denominator are there in a small thickening of M? Under what conditions can we count them asymptotically as the size of the denominator goes to infinity? We discuss some recent breakthrough result of Huang in this direction as well as a generalization for certain families of manifolds in higher codimension in work of the speaker and Shuntaro Yamagishi. We show that under certain curvature conditions we obtain stronger results for manifolds in higher codimension than for hypersurfaces, and discuss relations to Serre's dimension growth conjecture.

We develop a two dimensional version of the delta symbol method and apply it to establish quantitative Hasse principle for a smooth pair of quadrics defined over Q defined over at least 10 variables. This is a joint work with Simon Myerson (Warwick) and Junxian Li (Bonn).

 For an abelian variety over a number field, a consequence of the Birch and Swinnerton-Dyer conjecture is the parity conjecture: the global root number agrees with the parity of the Mordell-Weil rank. Although this remains largely open, the p-parity conjecture, which replaces the Mordell-Weil rank with the p∞-Selmer rank for a prime p, has proven more amenable to study. In the case when p = 2, however, comparatively little is known away from dimensions 1 and 2. In large part, this is due to a phenomenon observed by Poonen and Stoll that the Shafarevich–Tate group of a principally polarised abelian variety can have order twice a square. I will discuss instances where this obstruction can be overcome, presenting results for Jacobians of hyperelliptic curves and for principally polarised abelian varieties after quadratic extension of the base field.  

Elliptic curves have played a central role in the development of algebraic number theory and there is an elegant theory of the endomorphisms of elliptic curves.  Generalising to the higher-dimensional analogues of elliptic curves, called abelian varieties, more complex phenomena occur.

When we consider abelian varieties varying in families, there are often only finitely many members of the family whose endomorphism ring is larger than the endomorphism ring of a generic member.  The Zilber-Pink conjecture, generalising the André-Oort conjecture, predicts precisely when this finiteness occurs.

In this talk, I will discuss some of the progress which has been made on the Zilber-Pink conjecture, including a result of Daw and myself about isogenies between elliptic curves in families.

The infamous Ramanujan tau-function is the starting point for many mysterious conjectures and difficult open problems within the realm of modular forms. In this talk, I will discuss some of our recent results pertaining to odd values of the Ramanujan tau-function. We use a combination of tools which include the Primitive Divisor Theorem of Bilu, Hanrot and Voutier, bounds for solutions to Thue–Mahler equations due to Bugeaud and Gyory, and the modular approach via Galois representations of Frey-Hellegouarch elliptic curves. This is joint work with Mike Bennett (UBC), Adela Gherga (Warwick) and Samir Siksek (Warwick).