DIAMANT-GQT: Arithmetic and geometry

Organisers: David Holmes (Leiden University), Martijn Kool (Utrecht University)

Christopher Lazda (University of Amsterdam)

Zeta functions, Weil-Deligne representations and \(l\)-independence for local fields

Any scheme of finite type over \(\mathbb{Z}\) has a well-defined zeta function; for example, the zeta function of \(\mathbb{Z}\) itself is the famous Riemann zeta function, but the theory also encompasses zeta functions of curves over finite fields, and much more exotic objects. In all these cases, all the local factors of the zeta function have a beautiful cohomological interpretation. Starting with such a scheme over \(\mathbb{Q}\), however, it is not a priori clear how to define its zeta function, since different models over \(\mathbb{Z}\) can and do have different zeta functions. One way around this is to use the cohomological interpretation of the local factors as a definition, but to ensure that this makes sense, one needs to know \(l\)-independence results for the cohomology of varieties over local fields. In my talk I will explain how to formulate these independence results in a very precise manner using the theory of Weil-Deligne representations, and explain a little bit how to prove them for smooth and proper varieties over local function fields.

Francesca Balestrieri (Max Planck Institute, Bonn, Germany)

Arithmetic of zero-cycles on products of Kummer varieties and K3 surfaces

The following is joint work with Rachel Newton. In the spirit of work by Yongqi Liang, we relate the arithmetic of rational points to that of zero-cycles for the class of Kummer varieties over number fields. In particular, if \(X\) is any Kummer variety over a number field \(k\), we show that if the Brauer-Manin obstruction is the only obstruction to the existence of rational points on \(X\) over all finite extensions of \(k\), then the Brauer-Manin obstruction is the only obstruction to the existence of a zero-cycle of any odd degree on \(X\). Building on this result and on some other recent results by Ieronymou, Skorobogatov and Zarhin, we further prove a similar Liang-type result for products of Kummer varieties and K3 surfaces over \(k\).