# Workshop on Cohomological Methods in Arithmetic Geometry

s10.09.2012-14.09.2012

**Organized by:** H. Esnault, A. Kresch, B. Poonen, A. Skorobogatov

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Tim Browning (Bristol): How frequently does the Hasse principle fail?

*Abstract:*Counter-examples to the Hasse principle are known for many families of geometrically rational varieties. We discuss how often such failures arise for Chatelet surfaces and, if time permits, for certain higher-dimensional hypersurfaces. This is joint work with Regis de la Breteche.

Ulrich Derenthal (LMU München): Values of quadratic polynomials represented by norms

*Abstract:*Let K/k be an extension of number fields. When can values of a quadratic polynomial P(t) over k be represented by norms of elements of K? To answer this question, we will discuss the Hasse principle and weak approximation for the affine variety defined by P(t)=Norm

_{K/k}(x), in particular in the case [K:k]=4. This is joint work with A. Smeets and D. Wei, extending recent work of T. D. Browning and D. R. Heath-Brown.

Gerd Faltings (MPI Bonn): Mathematics around Kim's method

*Abstract:*I give the background from the theory of crystalline Galois-representations used in Kim's proof of finiteness of integral points.

Wojtek Gajda (AMU Poznań): Independence of l-adic representations over function fields

David Harari (Orsay): The unramified Brauer group of a homogeneous space

*Abstract:*This is joint work with C. Demarche and M. Borovoi. We compute the unramified Brauer group of certain homogeneous spaces over various fields (algebraically closed, local, global, finite), using arithmetical methods.

Yonatan Harpaz (Hebrew University Jerusalem): The Relative Étale Shape and Rational Points

*Abstract:*In this talk we will describe some on going work relating homotopy theoretic properties of schemes and varieties to the study of rational points on them. We will introduce a generalization of the étale homotopy type of Artin and Mazur to a relative setting X → S and explain how one can apply homotopy theory to it in order to study S-rational points of X. In the case where S is the spectrum of a number field the theory can be used to obtain a unified view of classical arithmetic obstructions such as the Brauer-Manin obstruction and descent obstructions. This is joint work with Tomer Schlank.

Nick Katz (Princeton): TBA

Yongqi Liang (Orsay): Brauer-Manin obstruction for zero-cycles on rationally connected varieties defined over number fields

*Abstract:*In this talk, we consider rationally connected varieties defined over number fields. We will state a general relation between the local-global principle for rational points and for zero-cycles. As an application, we prove the exactness of a sequence of local-global type for homogeneous spaces of linear algebraic groups.

Max Lieblich (University of Washington): TBA

Dino Lorenzini (University of Georgia): An Avoidance Lemma and a Moving Lemma for families

*Abstract:*Given a projective morphism X→S over an affine base, I'll discuss a technique for proving the existence of hypersurfaces H in X with various favorable properties. Applications, for some classes of X→S, include an Avoidance lemma in families, the existence of finite quasi-sections for X→S, the existence of a finite morphism from X to a projective n-space over S, and a moving lemma for horizontal 1-cycles. This is joint work with O. Gabber and Q. Liu.

Lilian Matthiesen (Bristol): Rational points on conic bundle surfaces via additive combinatorics (joint work with Tim Browning and Alexei Skorobogatov)

Ambrus Pál (Imperial College London): TBA

Jakob Stix (Heidelberg): On the birational section conjecture with local conditions

*Abstract:*Grothendieck's section conjecture predicts a description of rational points of a hyperbolic curve entirely in terms of profinite (fundamental) groups. We will present and prove a version of the conjecture that assumes additional mainly local properties for the sections involved.

Bianca Viray (Brown University): Vertical Brauer elements and del Pezzo surfaces of degree 4

Olivier Wittenberg (ENS Paris): Divisibility of Chow groups of 0-cycles of varieties over strictly local fields

*Abstract:*Let X be a smooth projective variety defined over the maximal unramified extension of a p-adic field, or over k((t)) with k algebraically closed of characteristic p. S. Saito and K. Sato have proved that the Chow group of zero-cycles of degree 0 on X up to rational equivalence is the direct sum of a finite group and a group which is divisible by any prime different from p. We study this finite group and show that it vanishes for simply connected surfaces with geometric genus zero as well as for K3 surfaces with semi-stable reduction if p=0, but that it does not vanish for arbitrary simply connected surfaces. In particular, the cycle class map with finite (prime to p) coefficients need not be injective. (Joint work with H. Esnault.)

Trevor Wooley (Bristol): Weak approximation for rational morphisms from the projective line to a diagonal hypersurface

*Abstract:*We investigate the space of rational morphisms from the projective line to a hypersurface defined by a diagonal equation of degree d in n+1 variables. Provided that n is large enough in terms of d and the degree of the rational morphism, one can establish a weak approximation property. It transpires that recent work on "efficient congruencing" shows that "large enough" is not particularly large. This is joint work with Scott Parsell and Sean Prendiville.