Talks

I am an honest salesman - I will talk about arithmetic and geometric reciprocity laws. Familiarity with quadratic reciprocity and the fundamental theorem of calculus would be helpful, but both will be recalled during the talk.

A bit unoriginally, motivated by OpenAI's recent 'bulk release', in this talk we will explore in more detail the connections between the (generalised) Riemann hypothesis and certain problems in number theory. We will start with a gentle introduction to the Riemann zeta function, its zeros, and how they relate to the distribution of rational primes. Along the way, we will also discuss some of its generalisations, particularly Dedekind zeta functions and Hecke \(L\)-functions over arbitrary number fields. The main part of the talk will concern the quasi-Riemann hypotheses and their consequences, focusing on situations in which the full generalised Riemann hypothesis can be replaced by a weaker one. In particular, we will discuss conditional effective versions of the Chebotarev density theorem and some (more or less) arithmetic applications of these results. If time permits, we will also briefly outline some of the main ideas behind the AI's approach in OpenAI's manuscript on the zero-free half-plane \(\operatorname{Re}(s)>11/12\) for the Riemann zeta function \(\zeta(s)\).

The Langlands Program is often presented as a complex web of conjectures so subtle that coming up with a precise conjecture is already a very difficult task. This is true, but we can try to make it less scary. In this mainly non-formal talk I will try to take you on a sightseeing tour of this web. I will introduce some of the main questions of the Program as well as giving the existing results along the way.

The study of Diophantine equations is a classical and notoriously difficult problem. Perhaps the most fundamental question one can ask is that of the existence of a solution over a global field \(K\). A necessary condition is local solubility – the existence of solutions over each completion of \(K\). When this is in fact sufficient, we say that the Hasse principle holds. This is true for example of quadratic forms, but fails in general. Historically, such failures were established using various ad hoc methods, but there was no approach that could be applied in general. This changed in the '70s and '80s with the introduction of the unifying frameworks of the Brauer–Manin obstruction and the theory of descent by torsors, each of which heavily exploit the geometry of the underlying varieties. We shall motivate and discuss both of these approaches with plenty of examples, explaining how they respectively generalise classical methods of applying abelian reciprocity laws and of descent on elliptic curves (which in turn generalises Fermat's infinite descent). Finally, we will see how these two different frameworks turn out to be deeply related.

The conjecture of Birch and Swinnerton-Dyer claims that the order of vanishing \(r\) of the Hasse–Weil \(L\)-function of an elliptic curve \(E\) over a number field \(K\) at \(s=1\) is equal to its own rank. In this talk we describe how one can use the theory of Euler systems applied to Kato's zeta elements to prove this claim in the case where \(K=\mathbb{Q}\), \(r=0\), and \(E\) does not have complex multiplication.