Arithmetic geometry of K3 surfaces and related areas

Imperial College London, 3-4 October 2024

Workshop organised by Oliver Gregory, Christian Liedtke, Alexei Skorobogatov and funded by EPSRC, Heilbronn Institute for Mathematical Research, and the partnership between Technical University of Munich and Imperial College London

Thursday 3 October

Roderic Hill Building RODH 414

9.30-10.30 Christian Liedtke Torsors over pointed singularities

11.00-12.00 Damián Gvirtz-Chen Non-thin rational points for doubly elliptic K3 surfaces

2.00-3.00 Rachel Newton Transcendental Brauer-Manin obstructions on singular K3 surfaces

3.30-4.30 Domenico Valloni Linear Brauer classes in positive characteristic

4.30-5.00 Open problems/discussion

Friday 4 October

Roderic Hill Building RODH 422

9.30-10.30 Thomas Geisser Brauer and Neron-Severi groups of surfaces over finite fields

11.00-12.00 Oliver Gregory Griffiths groups in positive characteristic

2.00-3.00 Livia Grammatica Comparison of flat and crystalline cohomology via quasi-syntomic descent

3.15-4.15 Yuan Yang Remarks on the p-primary torsion of the Brauer group

4.30-5.30 Christopher Lazda Boundedness of the p-primary torsion of the Brauer groups of K3 surfaces

The workshop will take place in the Roderic Hill Building on Imperial’s South Kensington campus. The nearest tube stations are Gloucester Road and South Kensington. Oli will be at the Prince Consort Road entrance to swipe you through the barriers. Hopefully there will be a guard at the gate who I can delegate that duty to let participants in throughout the day. Note that the campus map shows that the Roderic Hill Building can be accessed through the ACEX building. I do not know this is a way to access Roderic Hill without needing a swipe card.

A link to the campus map: https://www.imperial.ac.uk/media/imperial-college/visit/public/SouthKensingtonCampus.pdf

Abstracts

Christian Liedtke

Let G be a finite group scheme over some algebraically closed field k and assume that it acts on (completed) affine n-space A^n such that the origin is the only fixed point. Let X = A^n/G be the associated quotient space and let x \in X be the image of the origin. In the case where k is the field of complex numbers, then G is a finite group, one may assume w.l.o.g. that the G-action is linear, and the fundamental group of X-{x} recovers both G and its action on A^n. If k is algebraically closed of positive characteristic and G is linearly reductive, this is still true, but the proofs are much harder. Finally, if k is algebraically closed of positive characteristic and G is not linearly reductive, then things become complicated, subtle, and pathological. This is joint work with Gebhard Martin and Yuya Matsumoto.

Damián Gvirtz-Chen

We prove that elliptic K3 surfaces over a number field which admit a second elliptic fibration satisfy the potential Hilbert property. Equivalently, the set of their rational points is not thin after a finite extension of the base field. Furthermore, we classify those families of elliptic K3 surfaces over an algebraically closed field which do not admit a second elliptic fibration. Joint work with G. Mezzedimi.

Rachel Newton

Let E and E′ be elliptic curves over Q with complex multiplication by the ring of integers of an imaginary quadratic field K and let Y = Kum(E×E′) be the minimal desingularisation of the quotient of E×E′ by the action of −1. We study the Brauer groups of such surfaces Y and use them to furnish new examples of transcendental Brauer–Manin obstructions to weak approximation. This is joint work with Mohamed Alaa Tawfik.

Domenico Valloni

I will define linear Brauer classes in positive characteristic and explain their role in the Brauer-Manin obstruction. I will also characterize the Brauer-Manin set cut out by p-torsion Brauer classes for varieties with many differential forms, e.g., curves of genus greater than 2. Finally, I will talk about the existence of linear Brauer classes for supersingular K3 surfaces. This is joint work with Alexei Skorobogatov. 

Thomas Geisser

For a smooth and proper surface over a finite field, the formula of Artin and Tate relates the behavior of the zeta-function at 1 to other invariants of the surface. We give a version which equates invariants only depending on the Brauer group to invariants only depending on the Neron-Severi group. We estimate of the terms appearing in the formula and discuss the special case of abelian varieties and K3-surfaces. 

Oliver Gregory

The n-th Griffiths group of a variety X is an invariant which measures the difference between homological equivalence and algebraic equivalence for codimension n cycles in X. I will first recall how to think about Griffiths groups and how they behave in characteristic zero. I will then discuss some new results and open problems in characteristic p>0. 

Livia Grammatica

Let X be a smooth scheme over an algebraically closed field of characteristic p. A classical result of Illusie relates the fppf cohomology of X with coefficients Z_p(1) and the slope 1 part of its integral crystalline cohomology. The original proof uses the theory of the de Rham-Witt complex, which relies heavily on differential forms in characteristic p with the Cartier isomorphism being a fundamental ingredient. In this talk we present a different proof, stemming from work of Scholze, Bhatt, Lurie and others, where differential forms are absent, and the argument is centred around cohomological descent for the coperfection map from X_perf to X.

Yuan Yang

Let X be a smooth projective variety over an algebraically closed field k of positive characteristic p. The Brauer group Br(X) is a direct sum of finitely many copies of Q_p/Z_p and an abelian group of finite exponent. The latter is an extension of a finite group J by the group of k-points of a connected commutative unipotent algebraic group U. We show that (1) if X is ordinary, then U = 0; (2) if X is a surface, then J is the Pontryagin dual of NS(X)[p^∞]; (3) if X is an abelian variety, then J = 0, and we compute the dimension of U using Crew’s formula. We also compute Br(X)[p^∞], where X is an Enriques surface.

Christopher Lazda

The transcendental Brauer group of a variety X over a field k is the image of its Brauer group inside the Brauer group of the base change of X to a separable closure of k. If X is a K3 surface, and k is finitely generated of characteristic 0, then it was shown by Skorobogatov and Zarhin that this group is finite. If k is finitely generated of characteristic p (and X is again a K3 surface), then later work of Skorobogatov and Zarhin (in the case p =/=2) and Ito (in the case p=2) showed that its prime-to-p torsion subgroup is finite. One cannot in general expect finiteness of p-torsion in characteristic p, however, I will explain how to use Madapusi-Pera’s proof of the Tate conjecture for K3 surface to show that one does have such a finiteness result in the case that X is non-supersingular. Combined with known results in the supersingular case, this shows that in general, the p-torsion will always at least be of finite exponent, that is, annihilated by a fixed power of p. This is joint work with Alexei Skorobogatov.