Monday 9th September 2024: University of Oxford

Aleksander Horawa

Special values of L-functions

In 1735, Euler observed that ζ(2) = 1 + 1/2² + 1/3² + ⋯ = π²/6, which is related to the famous identity ζ(−1) "=" 1 + 2 + 3 + ⋯ "=" −1/12. In general, values of the Riemann zeta function at positive even integers are rational numbers multiplied by a power of π. The values at positive odd integers are much more mysterious; for example, Apéry proved that ζ(3) = 1 + 1/2³ + 1/3³ + ⋯ is irrational, but we still don't know if ζ(5) = 1 + 1/2⁵ + 1/3⁵ + ⋯ is rational or not! In this talk, we will explain the arithmetic significance of these values, their generalizations to Dirichlet/Dedekind L−functions, and to L−functions of elliptic curves. We will also present a new formula for ζ(3) = 1 + 1/2³ + 1/3³ + ... in terms of higher algebraic cycles which came out of an ongoing project with Lambert A'Campo.

Lilian Matthiesen

Distributional properties of smooth numbers: Orthogonality to nilsequences and beyond

An integer is called y-smooth if all of its prime factors are of size at most y. The y-smooth numbers below x form a subset of the integers below x which is, in general, sparse but is known to enjoy good equidistribution properties in progressions and short intervals. Distributional properties of y-smooth numbers found striking applications in, for instance, integer factorisation algorithms or in work of Vaughan and Wooley on improving bounds in Waring's problem.

In the first part of this talk I will discuss joint work with Mengdi Wang which considers some finer aspects of the distribution of y-smooth numbers. More precisely, we show for a very large range of the parameter y that y-smooth number are (in a certain sense) discorrelated with `nilsequences'. Through work of Green, Tao and Ziegler, our result is closely related to the Diophantine problem of studying solutions to certain systems of linear equations in the set of y-smooth numbers. In the second part, I will discuss new progress towards such problems concerning the full range of the smoothness parameter from before.

Adam Morgan

Hasse principle for intersections of two quadrics via Kummer surfaces

I will discuss recent work with Skorobogatov in which we establish the Hasse principle for a broad class of degree 4 del Pezzo surfaces (including all those with irreducible characteristic polynomial), conditional on finiteness of Tate--Shafarevich groups of abelian surfaces. A corollary of this work is that the Hasse principle holds for smooth complete intersections of two quadrics in P^n for n\geq 5, conditional on the same conjecture. This was previously known by work of Wittenberg assuming both finiteness of Tate--Shafarevich groups of elliptic curves and Schinzel's hypothesis (H). 

I will also discuss forthcoming work with Lyczak which, again under finiteness of relevant Tate--Shafarevich groups, shows that the Brauer--Manin obstruction explains all failures of the Hasse principle for certain degree 4 del Pezzo surfaces about which nothing was known previously.  

Jack Thorne

100% of odd hyperelliptic Jacobians have no rational points of small height

Reduction theory gives a way of reducing the coefficients of objects such as binary quadratic forms using the action of arithmetic groups such as SL_2(Z). I will discuss a surprising new connection between reduction theory and heights of rational points, that can be exploited to prove that, with probability 1, the Jacobian of an odd hyperelliptic curve has no non-trivial rational points of small height (in a precise sense). This is joint work with Jef Laga.