The London-Paris Number Theory Seminar meets twice per year, once in London and once in Paris. It is supported by a grant from the London Mathematical Society.
London organizers: David Burns, Kevin Buzzard Fred Diamond, Alexei Skorobogatov, Andrei Yafaev.
Paris organizers: Pierre Charollois, David Harari, Michael Harris, Marc Hindry, Benjamin Schraen, Jacques Tilouine.
The schedule:
Monday 3 June, pm
2-3 F. Charles. On the twisted Lang-Weil estimates.
3.30-4.30 Y. Harpaz. The section conjecture for graphs and applications for singular curves
5-6 J.-L. Colliot-Thélène. Espaces homogènes sur les corps de fonctions de courbes sur un corps local
Tuesday 4 June, am
10-11 O. Wittenberg. On the kernel of the cycle class map for 0-cycles over local fields
11.30-12.30 D. Testa. Computability of Néron-Severi groups
Abstracts:
François Charles.
Given an algebraic variety over a finite field, the classical Lang-Weil estimates compute the number of intersection point of the diagonal with the graph of Frobenius. Hrushovski has proven a twisted version of these estimates, replacing the diagonal by a more general correspondence. While Hrushovski's proof relies on non-classical objects such as difference schemes, this has proven to yield a number of applications to algebraic geometry. In this talk, we will explain a self-contained intersection-theoretic proof of the twisted Lang-Weil estimates. This is joint work with Damien Rössler.
J.-L. Colliot-Thélène.
Over such a function field F, D. Harbater, J. Hartmann and D. Krashen have proved a local-global principle for the existence of rational points on principal homogeneous spaces under a connected linear algebraic group G over F when the underlying variety of G is F-rational, i.e. birational to affine space over the field F. In recent work with Parimala and Suresh, we show that this local-global principle may fail when the group G is not F-rational. The obstruction we use comes from the Bloch-Ogus complex for étale cohomology over an arithmetic surface extending the curve. One may then ask when this new obstruction is the only obstruction to the existence of rational points.
Y. Harpaz.
The connection between rational points and sections of the Grothendieck exact sequence has an analogue for topological spaces carrying an action of a finite group. The corresponding section conjecture in this setting is false in general (as for general varieties), but is true when the underlying space is a graph. In this lecture we will explain this result and show how it can be applied to prove that finite descent is the only obstruction to the Hasse principle for a certain class of singular curves.
D. Testa.
The Néron-Severi group of a variety is a finitely generated abelian group whose rank is the Picard number of the variety itself; it plays an important role in the determination of the algebraic part of the Brauer-Manin obstruction. I will begin with an introduction to Néron-Severi groups, and will discuss a few methods that have been used to determine Picard numbers in special cases. Then I will report on recent joint work with Bjorn Poonen and Ronald van Luijk on the computability of Picard numbers in general.
O. Wittenberg.
I will discuss the kernel of the cycle class map from the Chow group of 0-cycles to etale cohomology with finite or integral coefficients, in the case of surfaces defined over a p-adic field, including surfaces with nonzero geometric genus. (Joint work with H. Esnault.)
This page is maintained by Kevin Buzzard.