# 3rd Workshop on Interactions between Arithmetic and Homotopy

## 14th-16th February, 2017 Imperial College London

Talks and coffee breaks will be at 170 Queen's Gate, in the Drawing Room.

 Tues 14 9:30-10:00 Registration 10:00-11:00 Michel Gros On Hodge-Tate local systems 11:00-11:30 Coffee 11:30-12:30 Fréderic Déglise $$p$$-adic Hodge theory in motivic homotopy 12:30-14:30 Lunch 14:30-15:30 Moshe Jarden Torsion on abelian varieties over large algebraic extensions of finitely generated extensions of $$\mathbb Q$$ 15:30-16:00 Coffee 16:00-17:00 Alex Betts Non-abelian Bloch-Kato Selmer sets - a cosimplicial approach

 Wed 15 10:00-11:00 Charles Vial Distinguished models of intermediate Jacobians 11:00-11:30 Coffee 11:30-12:30 Chris Lazda A semistable Lefschetz $$(1,1)$$-theorem in equicharacteristic 12:30-14:30 Lunch 14:30-15:30 Ido Efrat Massey products of absolute Galois groups 15:30-16:00 Coffee 16:00-17:00 Kirsten Wickelgren Enriched Euler numbers and an arithmetic count of the lines on a cubic surface 19:30- Conference Dinner

 Thur 17 10:00-11:00 Olivier Wittenberg Fourfold Massey products over number fields 11:00-11:30 Coffee 11:30-12:30 Ambrus Pál Vanishing of Massey products and computations in motivic cohomology 12:30-14:30 Lunch 14:30-15:30 Jonathan Pridham Smooth functions on algebraic $$K$$-theory 15:30-16:00 Coffee 16:00-17:00 Netan Dogra An effective Chabauty-Kim Theorem

Speaker: Alex Betts

Title: Non-abelian Bloch-Kato Selmer sets - a cosimplicial approach

Abstract: Non-abelian analogues of local Bloch-Kato Selmer groups appear in Minhyong Kim's anabelian proof of Siegel's theorem, where they play an important role in the study of integral points via cohomological methods. In this talk, we will demonstrate how tools from homotopical algebra allow one to study these non-abelian Bloch-Kato Selmer sets in the abstract, parallel to the homological-algebraic tools which one uses in the abelian setting. As an example, we examine the Bloch-Kato Selmer sets obtained from the unipotent fundamental groups of $$\mathbb G_m$$-torsors on abelian varieties, and discuss how they allow one to recover local components of height functions.

Speaker: Fréderic Déglise

Title: $$p$$-adic Hodge theory in motivic homotopy

Abstract: I will present a work in collaboration with Wiesia Niziol which aims to incorporate $$p$$-adic Hodge theory into the framework of modules over ring spectra, in the sense of Morel-Voevodsky's motivic homotopy theory. Our main result is the identification of "modules over syntomic cohomology" as a full subcategory of the derived category of potentially semi-stable representations, making use of ideas of Beilinson and Drew. I will then present an ongoing project to extend Fontaine semi-stable comparison to a suitable notion of syntomic modules. The later should be compared to Saito mixed Hodge modules, and our objective is to get some kind of $$p$$-adic Riemann-Hilbert correspondence.

Speaker: Netan Dogra

Title: An effective Chabauty-Kim Theorem

Abstract: In a series of papers Kim showed how the unipotent fundamental group of a curve of genus bigger than 1 can be used to study the set of rational points, generalising Chabauty's work from the 1930s. In this talk I will explain how the "Chabauty-Kim sets" obtained by this theory can be described explictly, and can in certain cases be used to bound the number of the rational points of the curve, generalising Coleman's effective Chabauty theorem. This is joint work with Jennifer Balakrishnan.

Speaker: Ido Efrat

Title: Massey products of absolute Galois groups

Abstract: Given a field containing a $$p$$-th root of unity ($$p$$ prime), one is interested in the cohomology ring of its absolute Galois groups, with $$\mathbb F_p$$-coefficients. It structure as a graded algebra with cup product is well described by the celebrated Voevodsky-Rost theorem: namely, it is the Milnor K-ring of the field modulo $$p$$. However, the cohomology algebra is also equipped with external operations, such as the Massey products. We will survey the extensive work done over the past few years to understand this external structure.

Speaker: Michel Gros

Title: On Hodge-Tate local systems

Abstract: We will explain the role of Hodge-Tate local systems from the perspective of the $$p$$-adic Simpson correspondence (joint work with A. Abbes).

Speaker: Moshe Jarden

Title: Torsion on abelian varieties over large algebraic extensions of finitely generated extensions of $$\mathbb Q$$

Abstract: Let $$K$$ be a finitely generated extension of $$\mathbb Q$$ and $$A$$ a non-zero abelian variety over $$K$$. Let $$\overline K$$ be the algebraic closure of $$K$$ and Gal$$(K)= \textrm{Gal}(\overline K/K)$$ the absolute Galois group of $$K$$ equipped with its Haar measure. For each $$\sigma\in\textrm{Gal}(K)$$ let $$\overline K(\sigma)$$ be the fixed field of $$\sigma$$ in $$\overline K$$. We prove that for almost all $$\sigma\in\textrm{Gal}(K)$$ there exist infinitely many prime numbers $$l$$ such that $$A_l(\overline K(\sigma))\neq0$$. This completes the proof of a conjecture of Geyer-Jarden from 1978 in characteristic $$0$$. Joint work with Sebastian Petersen.

Speaker: Chris Lazda

Title: A semistable Lefschetz $$(1,1)$$-theorem in equicharacteristic

Abstract: By using some elementary properties of the logarithmic de Rham-Witt I will explain how to prove that a rational line bundle on the special fibre of a proper, semistable scheme over a power series ring $$k[[t]]$$ in characteristic $$p$$ lifts to the total space if and only if its first Chern class does. This generalises a result of Morrow in the smooth case, and provides an equicharacteristic analogue of a result of Yamashita. I will also explain a corollary concerning algebraicity of cohomology classes on varieties over global function fields. This is joint work with Ambrus Pál.

Speaker: Ambrus Pál

Title: Vanishing of Massey products and computations in motivic cohomology

Abstract: The most general results on the vanishing of higher Massey products in Galois cohomology use the Bloch-Kato conjectures. However it is possible to give very elegant and short proofs of such vanishing results in certain cases by working with motivic cohomology directly. (Joint work with Tomer Schlank.)

Speaker: Jonathan Pridham

Title: Smooth functions on algebraic $$K$$-theory

Abstract: For any complex scheme $$X$$, we can define smooth functions on algebraic $$K$$-theory by derived Kan extension. For $$X$$ smooth and proper, it turns out that the resulting complex is almost dual to real Deligne cohomology. Thus Beilinson's regulator can be interpreted as a map from points to compactly supported distributions.

Speaker: Charles Vial

Title: Distinguished models of intermediate Jacobians

Abstract: Let $$X$$ be a smooth projective variety defined over a subfield $$K$$ of the complex numbers. It is natural to ask whether the complex abelian variety that is the image of the Abel-Jacobi map defined on algebraically trivial cycles admits a model over $$K$$. I will show that it admits a unique model making the Abel-Jacobi map equivariant with respect to the action of the automorphism group of the complex numbers fixing $$K$$. As an application, we answer a question of Mazur: we show that this model over the base field $$K$$ is dominated by the Albanese variety of a product of components of the Hilbert scheme of $$X$$. We also recover a result of Deligne on complete intersections of Hodge level one. This is joint work with Jeff Achter and Yano Casalaina-Martin.

Speaker: Kirsten Wickelgren

Title: Enriched Euler numbers and an arithmetic count of the lines on a cubic surface

Abstract: A celebrated 19th century result of Cayley and Salmon is that a smooth cubic surface over the complex numbers contains exactly 27 lines. Over the real numbers, the number of lines depends on the surface, but Segre showed that a certain signed count is always 3. We extend this count to an arbitrary field using $$\mathbb A^1$$-homotopy theory: we define an Euler number in the Grothendieck-Witt group for a relatively oriented algebraic vector bundle as a sum of local degrees, and then generalize the count of lines to a cubic surface over an arbitrary field. This is joint work with Jesse Leo Kass.

Speaker: Olivier Wittenberg

Title: Fourfold Massey products over number fields

Abstract: Massey products in the Galois cohomology of an arbitrary field are conjectured to always vanish. This conjecture is known to hold for triple Massey products. We establish it for fourfold Massey products in the Galois cohomology of number fields with coefficients in $$\mathbb Z/2\mathbb Z$$. (Joint work with Pierre Guillot, Jan Mináč, Adam Topaz.)