Schedule of the Research Conference

Victor Abrashkin (University of Durham): p-extensions of local fields with Galois groups of nilpotence class less than p

Nilpotent analogue of the Artin-Schreier theory was developed by the author about twenty years ago. It has found already applications in an explicit description of the ramification filtration modulo p-th commutators and the proof of analogue of the Grothendieck Conjecture for local fields. We remind basic constructions of this theory and indicate further progress in the study of local fields of mixed characteristic, especially, higher dimensional local fields.

Yuri Bilu (IMB Université Bordeaux I): Integral points on modular curves

The problem of determination of rational points on modular curves reduces grosso mode to three types of curves of prime level, corresponding to three types of maximal subgroups of the linear group GL₂(F_p):

• the curve X₀(p), corresponding to the Borel subgroup;
• the curve X_sp⁺(p), corresponding to the normalizer of a split Cartan subgroup;
• the curve X_ns⁺(p), corresponding to the normalizer of a non-split Cartan subgroup.

On the curves of the first two types rational points are determined (almost) completely: Mazur (1978), B.-Parent-Rebolledo (2012). In particular, it is proved that for p>13 the rational points are either cusps or the CM-points.  Little is known however on rational points on X_ns⁺(p).
I will speak on a recent progress in a simpler problem: classification of integral points on X_ns⁺(p) (i.e. rational points P such that j(P) is in Z). My students Bajolet и Sha obtained a rather sharp upper bound for the size of integral points. Also, in a joint work with Bajolet we proved that for 7<p<71 there are no integral points on X_ns⁺(p) other than the CM-points; this improves on a recent work of Schoof and Tzanakis, who proved this for p=11.

Tim Browning, Lilian Matthiesen (University of Bristol): Norm forms as products of linear polynomials

We report on recent progress using additive combinatorics to prove the Hasse principle and weak approximation for certain varieties defined by systems of equations involving norm forms. This is used to show that the Brauer-Manin obstruction controls weak approximation on normic bundles of the shape N_K(x₁,...,x_n) = P(t), where P(t) is a product of linear polynomials all defined over the rationals and K is an arbitrary degree n extension of the rationals.

Jörg Brudern (Universität Göttingen): Random diophantine equations

We address classical questions concerning diagonal forms with integer coefficients. Does the Hasse principle hold? If there are solutions, how many? If there are solutions, what is the size of the smallest solutions? In joint work with Dietmann, nearly optimal answers to such questions were obtained for almost all forms (in the sense typically attributed to "almost all" in the analytic theory of numbers) provided that the number of variables exceeds three times the degree of the forms under consideration.

Przemyslaw Chojecki (Institut Mathématique de Jussieu): On mod p non-abelian Lubin-Tate theory for GL(2)

Non-abelian Lubin-Tate theory for GL(2) describes the l-adic cohomology of the Lubin-Tate tower for GL(2,Q_p) in terms of the Langlands program. Until recently, all results were stated under the assumption that p is different from l. We discuss the case when l=p and we give a partial description of the mod p etale cohomology of the Lubin-Tate tower in terms of the mod p local Langlands correspondence.

Jean-Louis Colliot-Thélène (Université Paris-Sud): Strong approximation in a family

Let a_i(t), i=1,2,3, and p(t) be polynomials in Z[t]. Assume the product p(t) ⋅ ∏ a_i(t) is not constant and has no square factor. Consider the equation ∑ a_i(t) x_i² = p(t). Assume that for all t ∈ R the real conic ∑ a_i(t) x_i² = 0 has point over R.
Then strong approximation holds for the integral solutions of this equation. The set of solutions with coordinates in Z is dense in the product of integral local solutions at all finite primes. In particular, there is a local-global principle for integral points.
The case where all the a_i(t) are constant was dealt with in an earlier paper with F. Xu (Beijing). The result above is itself a special case of a general theorem on families of homogenous spaces of semisimple groups, obtained jointly with D. Harari (Paris-Sud).

Ulrich Derenthal (Universität München): Counting points over imaginary quadratic number fields

For Fano varieties over number fields, the distribution of rational points is predicted by Manin's conjecture. One approach uses universal torsors and Cox rings; over the field Q of rational numbers, this was started by Salberger for toric varieties and was extensively studied for many examples of del Pezzo surfaces. In this talk, I present an extension of this approach to imaginary quadratic number fields, in particular for some singular quartic del Pezzo surfaces (joint work with C. Frei).

David Rodney Heath-Brown (University of Oxford): Simultaneous representation of pairs of integers by quadratic forms

This is joint work with Lillian Pierce.  We examine the simultaneous integer equations Q₁(x₁,…,x_n) = m₁ and Q₂(x₁,…,x_n) = m₂, and show, under a smoothness condition, that "almost all" pairs m₁,m₂ which have local representations also have a global solution, as soon as n is at least 5. One can use this to attack Q₁(x₁,…,x_n) = Q₂(x₁,…,x_n) = 0 when n is at least 10. The proof uses a 2-dimensional circle method with a Kloostermann refinement.

Oscar Marmon (Georg-August-Universität Göttingen): The density of twins of k-free numbers

For k ≥ 2, we consider the number A_k(Z) of positive integers n ≤ Z such that both n and n+1 are k-free. In joint work with Dietmann, we prove an asymptotic formula A_k(Z) = c_k Z + O(Z^14/(9k)+ε), where the error term improves upon previously known estimates. The main tool used is the approximative determinant method of Heath-Brown.

David McKinnon (University of Waterloo): Approximating points on varieties

Famous theorems of Roth and Liouville give bounds on how well algebraic numbers can be approximated by rational numbers. In my talk, I will describe not-yet-famous generalizations of these theorems to arbitrary algebraic varieties, giving bounds on how well algebraic points can be approximated by rational points. This is joint work with Mike Roth of Queen's University, who is not yet related to famous-theorem Roth.

Ambrus Pal (Imperial College London): New two-dimensional counter-examples to the local-global principle

In my talk I will describe some new two-dimensional counter-examples to the local-global principle.

Florian Pop (University of Pennsylvania): Local-global principles for rational points

One of the most exciting developments originating from capacity theory (Fekete-Szego, Robinson, Cantor, Rumely) is Rumely's local-global principle, and its applications to the solvability of incomplete global Skolem problems (Cantor-Roquette, Roquette, and Moret-Bailly). My talk will be about complete Skolem problems and how they relate to some recent "complete capacity theory" results for curves developed by Rumely.

Mohamed Saidi (Exeter University): On the anabelian section conjecture over finitely generated fields

After introducing the section conjecture and some basic facts I will discuss in the first part of the talk a conditional result on the section conjecture over number fields. In the second part of the talk I will discuss the following result: the section conjecture holds over all finitely generated fields if it holds over all number fields, under the condition of finiteness of suitable Tate-Shafarevich groups.

Per Salberger (Chalmers University of Technology): Heath-Brown's determinant method and Mumford's geometric invariant theory

Heath-Brown's p-adic determinant method is used to count rational points on hypersurfaces and was extended to subvarieties of codimension >1 in projective space by Broberg and the author. The determinants that occur give rise to embeddings of Hilbert schemes in Grassmannians, which enables the use of techniques from geometric invariant theory. We describe some Diophantine applications and an unexpected link to the theory of Donaldson and Tian on Kähler metrics of constant scalar curvature.

Damaris Schindler (University of Bristol): Manin's conjecture for certain smooth hypersurfaces in biprojective space

So far, the circle method has been a very useful tool to prove many cases of Manin's conjecture. Work of B. Birch back in 1961 establishes this for smooth complete intersections in projective space as soon as the number of variables is large enough depending on the degree and number of equations. In this talk we are interested in subvarieties of biprojective space. There is not much known so far, unless the underlying polynomials are of bidegree (1,1). In this talk we present recent work which combines the circle method with the generalised hyperbola method developed by V. Blomer and J. Bruedern. This allows us to verify Manin's conjecture for certain smooth hypersurfaces in biprojective space of general bidegree.

Alexei Skorobogatov (Imperial College London): Applications of additive combinatorics to rational points

This is a joint work with Y. Harpaz and O. Wittenberg. In 1982 Colliot-Thélène and Sansuc noticed that Schinzel's Hypothesis (H) used in a fibration method going back to Hasse, has strong implications for local-to-global principles for rational points. In the case of the ground field Q when the degenerate fibres are all defined over Q, we show that in their method Hypothesis (H) can be replaced by the finite complexity case of the generalised Hardy-Littlewood conjecture, which is a recent theorem of Greeen, Tao and Ziegler. We sketch some of the applications of this observation.

Jeanine Marie Van Order (EPFL): Stable Galois averages of Rankin-Selberg L-values and nontriviality of p-adic L-functions.

I will present the notion of a stable Galois average of central values of GL(2) Rankin-Selberg L-functions, and then explain how to derive some consequences for the nonvanishing of certain families of p-adic L-functions (with interesting applications).

Gabor Wiese (Universite du Luxembourg): Symplectic Galois representations and applications to the inverse Galois problem

We give an account of joint work with Sara Arias-de-Reyna and Luis Dieulefait about compatible systems of symplectic Galois representations and how they can possibly be employed to the inverse Galois problem.
In the beginning of the talk the overall strategy will be outlined, starting from previous joint work with Dieulefait on the 2-dimensional case. We will then explain the existence of a minimal global field such that almost all the residual representations (of the compatible system) can be defined projectively over its residue fields. Moreover, we shall report on a very simple classification of symplectic representations containing a nontrivial transvection in their image. Finally, we shall combine the two points in an application to the inverse Galois problem.

Trevor Wooley (University of Bristol): Applications of efficient congruencing to rational points

We provide an overview of the implications of the new "efficient congruencing" method, first developed for Vinogradov's mean value theorem, so far as applications to problems involving rational points are concerned. Some of these conclusions are close to the convexity barriers in the circle method.