We iterate the algebraic étale-Brauer set of a nice variety $X$ over a number field $k$ and show that, when the geometric étale fundamental group of $X$ is finite, the iteration doesn't yield any new information. This result gives, among other things, evidence for the conjectures by Colliot-Thélène and Skorobogatov about the arithmetic behaviour of rational points on rationally connected varieties and K3 surfaces; moreover, it provides a partial answer to the algebraic version of a question by Poonen about iterating the descent set.Arithmetic aspects of the Burkhardt quartic, Nils Bruin (Simon Fraser University)
The Burkhardt quartic is a 3-dimensional projective hypersurface of degree 4 with many special properties that are classically known. For instance, its singular locus consists of 45 nodal singularities, and it has a particularly large projective automorphism group. It also has a modular interpretation: it parametrizes abelian surfaces with full level 3 structure. It is also known that the surface is birational to P^3 over the complex numbers.
We look at several arithmetic questions concerning the Burkhardt quartic: We show that the threefold is already rational over Q (Baker's original parametrization really required cube roots of unity). The exact parametrization also allows us to compute the zeta function of the various reductions (previously, this was only done for primes congruent to 1 mod 3).
We also look at the moduli interpretation and find an equation for an (almost) universal genus 2 curve with full level structure on its Jacobian, and find a particularly explicit description of how the level structure arises from the geometry of the Burkhardt quartic.
This is joint work with Brett Nasserden.The Manin constant in the semistable case, Kestutis Cesnavicius (University of California at Berkeley)
For an optimal modular parametrization J_0(n) --->> E of an elliptic curve E over Q of conductor n, Manin conjectured the agreement of two natural Z-lattices in the Q-vector space H^0(E, Omega^1). Multiple authors generalized his conjecture to higher dimensional newform quotients. We will discuss the semistable cases of the Manin conjecture and o f its generalizations using a technique that establishes general relations between the integral p-adic etale and de Rham cohomologies of abelian varieties over p-adic fields.Upper bounds for G-extensions over function fields, Jordan Ellenberg (University of Wisconsin at Madison)
We discuss recent joint work with TriThang Tran and Craig Westerland (https://arxiv.org/abs/1701.04541) in which we give an upper bound for the number of extensions of F_q(t) with fixed Galois group and bounded discriminant, which agrees with the bound conjectured by Malle. Such a bound is, for the moment, completely out of reach for number fields. The main new ingredient is an approach via quantum algebra to the cohomology of Hurwitz spaces.Galois action on homology of Fermat curves, Rachel Pries (Colorado State University)
We prove a result about the Galois module structure of the Fermat curve using commutative algebra, number theory, and algebraic topology. Specifically, we extend work of Anderson about the action of the absolute Galois group of a cyclotomic field on a relative homology group of the Fermat curve. By finding explicit formulae for this action, we determine the maps between several Galois cohomology groups which arise in connection with obstructions for rational points on the generalized Jacobian. Heisenberg extensions play a key role in the result. This is joint work with R. Davis, V. Stojanoska, and K. Wickelgren.Brauer groups and the Brauer-Manin sets of Kummer varieties, Alexei Skorobogatov (Imperial College)
This is a joint work with Yuri Zarhin. We study Kummer varieties attached to 2-coverings of abelian varieties of arbitrary dimension. Over a number field we show that the subgroup of odd order elements of the Brauer group does not obstruct the Hasse principle. Sufficient conditions for the triviality of the Brauer group are given, which allow us to give an example of a Kummer K3 surface of geometric Picard rank 17 over the rationals with trivial Brauer group. We establish the non-emptyness of the Brauer--Manin setof everywhere locally soluble Kummer varieties attached to 2-coverings of products of hyperelliptic Jacobians with large Galois action on 2-torsion.Cycles in the de Rham cohomology of abelian varieties over number fields, Yunqing Tang (Institute for Advanced Study)
In his 1982 paper, Ogus defined a class of cycles in the de Rham c ohomology of smooth proper varieties over number fields. In the case of abelian varieties, this class includes all the Hodge cycles by the work of Deligne, Ogus and Blasius. Ogus predicted that all such cycles are Hodge. In this talk, I will first introduce Ogus’ conjecture as a crystalline analogue of Mumford–Tate conjecture and explain how a theorem of Bost (using methods a la Chudnovsky) on algebraic foliation is related. After this, I will discuss the proof of Ogus’ conjecture for some families of abelian varieties under the assumption that the cycles lie in the Betti cohomology with real coefficients.Tropical geometry and uniformity of rational points on curves, David Zureick-Brown (Emory University)
Let X be a curve of genus g over a number field F of degree d = [F:Q]. The conjectural existence of a uniform bound N(g,d) on the number #X(F) of F-rational points of X is an outstanding open problem in arithmetic geometry, known to follow from the Bomberi-Lang conjecture. We prove a special case of this conjecture - we give an explicit uniform bound when X has Mordell-Weil rank r ≤ g- 3. This generalizes recent work of Stoll on uniform bounds on hyperelliptic curves. Using the same techniques, we give an explicit, unconditional uniform bound on the number of F-rational torsion points of J lying on the image of X under an Abel-Jacobi map. We also give an explicit uniform bound on the number of geometric torsion points of J lying on X when the reduction type of X is highly degenerate. Our methods combine Chabauty-Coleman's p-adic integration, non-Archimedean potential theory on Berkovich curves, and the theory of linear systems and divisors on metric graphs. This is joint work with Joe Rabinoff and Eric Katz.