Abstract: I will start by recalling the well-known structure of the Brauer group of a smooth proper variety over an algebraically closed field away from the p-primary torsion in characteristic p. Examples show that the p-primary torsion in characteristic p has more delicate nature, the case of abelian varieties being already almost completely open. The key part of the p-primary torsion of the Brauer group is a certain quasi-algebraic connected unipotent group. I will present a formula for its dimension and a sharp bound for its p-exponent, with examples for abelian varieties of small dimension. I will try to explain all necessary concepts as much as possible. The talk is based on joint work with Livia Grammatica and Yuan Yang.

Abstract: Mazur and Rubin proved that for every number field K, there exists an elliptic curve E defined over K with rank equal to zero. In this talk, we will explain how to construct for every number field K an elliptic curve E defined over K with rank equal to one. This is joint work with Carlo Pagano.

Abstract: For ℓ an odd prime number and d a squarefree integer, a central question in arithmetic statistics is to give pointwise bounds for the size of the ℓ-torsion of the class group of ℚ(√d). This is in general a difficult problem, and unconditional pointwise bounds are only available for ℓ = 3 due to work of Pierce, Helfgott–Venkatesh and Ellenberg–Venkatesh. The current record is h₃(d) ≪ε d^(1/3+ ε) due to Ellenberg–Venkatesh. We will discuss how to improve this to h₃(d) ≪ d³²⸍¹⁰⁰. This is joint work with Peter Koymans.

Abstract: For each square-free integer D, and each elliptic curve E, the 2-Selmer groups of E and its quadratic twist E_D naturally live in the same space. We are motivated to study their independence as E varies. We shall present a heuristic in this direction, and some results in support of it.

Abstract: I'll report on joint work with Shuntaro Yamagishi, which is directed at using function field analytic number theory to establish the irreducibility and dimension of the moduli space that parameterises rational surfaces of fixed degree contained in an arbitrary smooth hypersurface of small enough degree.

Abstract:  The local-global approach to the study of rational points on varieties over number fields begins by embedding the set of rational points on a variety X into the set of its adelic points. Then the Brauer—Manin pairing cuts out a subset of the adelic points that contains the closure of the rational points. Computing the Brauer—Manin pairing involves evaluating elements of the Brauer group of X at p-adic points. In this talk, I will discuss which primes are relevant for the Brauer—Manin obstruction, meaning that there is an element of the Brauer group with non-constant evaluation on p-adic points. I will present results from two collaborations: one with Martin Bright, and one with Emiliano Ambrosi and Margherita Pagano.

Abstract: In this talk, I will explain the relationship between Manin's conjecture over function fields and the geometric Manin conjecture over finite fields. In particular, I will discuss how to use results from the geometric side to obtain information on the arithmetic side with a special focus on del Pezzo surfaces and cubic hypersurfaces.

Abstract: Let X be a smooth projective hypersurface defined over Q. We provide new bounds for the cardinality of rational points of bounded height on X. If X is smooth and has degree at least 6, we improve the dimension growth conjecture bound. We achieve an analogous result for affine hypersurfaces whose projective closure is smooth.

Abstract: For n > 1, consider an absolutely irreducible polynomial F(Y, X₁, ..., Xₙ) that is a polynomial in Yᵐ and monic in Y. Let N(F, B) be the number of integral vectors x of height at most B such that there is an integral solution to F(Y, x) = 0. For m > 1 unconditionally, and m = 1 under GRH, we show that N(F, B) ≪ε log(∥F∥)ᶜ · B^(n-1+1/(n+1)+ε) under a non-degeneracy condition that encapsulates that F(Y, X₁, ..., Xₙ) is truly a polynomial in n + 1 variables. A strength of this result is that it requires no smoothness assumptions for F(Y, X₁, ..., Xₙ) nor constraints on the degrees of F in X₁, ..., Xₙ. A key ingredient in this work is a formulation of the Katz–Laumon stratification theorems for exponential sums that is uniform in families. This talk is based on joint work with Dante Bonolis, Emmanuel Kowalski, and Lillian B. Pierce.

Abstract: We present the main result of joint work with C. Frei, showing that 100% of diagonal conic bundle surfaces satisfy the Hasse principle. This is the first of two talks on this topic, focusing on a general 100%-result for averages of multivariate arithmetic functions over values of polynomials, obtained by Fourier-analytic methods.

Abstract: We present the main result of joint work with E. Sofos, showing that 100% of diagonal conic bundle surfaces satisfy the Hasse principle. This is the second of two talks on this topic, explaining how the general tool presented in the talk by E. Sofos is applied to obtain the main result.

Abstract: Let R be the field of real Puiseux series. It is a real closed field. In joint work with Alena Pirutka and Federico Scavia, we produce examples of smooth intersections of two quadrics in 5-dimensional projective space and in 9-dimensional projective space over R which are not stably rational but for which the space X(R) of R-points is semi-algebraically connected. The question of constructing such examples over the field of real numbers remains open. At the March 2025 Sofia meeting I had reported on the stable rationality problem for special threefolds fibered into quadrics over the line over the reals and any real closed field, after CT-Pirutka 2024 and Benoist-Pirutka 2024. This new work involves a semi-algebraic form of Ehresmann's theorem and a variation on the specialization method as extended in CT-Pirutka 2016.

Abstract: In this survey talk, we will explore various relations between the Galois cohomological properties and the Diophantine properties of fields. We will mainly focus on how rational points and zero-cycles behave on homogeneous spaces of linear algebraic groups and on low-degree hypersurfaces over fields with low cohomological dimension. Starting with some classical notions and results, we will finish by overviewing several recent results that I have obtained with Giancarlo Lucchini-Arteche in this context.

Abstract: The study of Campana points on varieties has gained a lot of attention in recent years. Philosophically, sets of Campana points on a variety over a number field interpolate between the set of rational points and the set of integral points. Several classical results and conjectures for rational points have been formulated also for Campana points. In this talk I will give an introduction to this topic, as well as talk about joint work with F. Balestrieri, J. Brandes, M. Kaesberg, J. Ortmann, and M. Pieropan, where we count Campana points of bounded height on non-linear diagonal hypersurfaces in projective space.

Abstract: I will report on work in progress with Julian Demeio and Rosa Winter dedicated to establishing the non-thinness of rational points in certain families of del Pezzo surfaces of degree one and Picard rank one, providing the first examples of the Hilbert property in this setting.

Abstract: In this talk, I will present a general framework for counting the number of rational points of bounded height symmetric squares of weak Fano varieties. I will then explain how this framework can be used to establish the Manin--Peyre conjecture for the symmetric square of an infinite family of non-split quadric surfaces. This is joint work with Francesca Balestrieri, Julian Lyczak, Jennifer Park and Nick Rome. 

Abstract: Localising points of bounded height is a way of refining weak approximation and is at the core of Manin's program. The geometric description of accumulating subsets and adelic measures provide a very precise prediction for the adelic distribution on almost Fano varieties. These predictions extend to the geometric setting leading to the understanding of the distribution of rational curves.

In its original form, this approach allows only to impose fixed conditions on a finite set of places, but it is natural to wish to push it beyond these limitations in two directions:

• Considering sets which depends on the bound on the height, in connection

with diophantine approximation

• Imposing conditions at all place, to be able to count, for example, curves which do not go through a given point.

The aim of this talk is to survey these extensions and their limitations. 

Abstract: Modular curves parametrize elliptic curves equipped with additional structure: the curve X₀(N) parametrizes elliptic curves with an N-isogeny, while X₁(N) parametrizes elliptic curves with a point of order N. Recently, there has been significant progress in studying the number of rational points of bounded height on modular curves. For example, for any N such that the modular curve X₀(N) has genus zero, the asymptotic behavior of the number of rational points of bounded naive height is known. However, these results cannot be explained by the standard Batyrev–Manin conjecture, due to the stacky structure arising from nontrivial automorphisms. We discuss how they fit within the framework of the Batyrev–Manin conjecture for stacks.

The talk is based on joint work with Changho Han and Mohammad Sadek.