15 novembre 2024 (Ecole Polytechnique, salle de séminaire du CMLS)

Si vous avez besoin d'aide pour atteindre la salle de séminaire, vous pouvez appeler Diego au 07 x 55 92 14, où x est l'ordre minimal d'un groupe simple non abélien.

11h: Michael Stoll (Université de Bayreuth).

Conjectural asymptotics of prime orders of points on elliptic curves over number fields

Define, for a positive integer $d$, $S(d)$ to be the set of all primes $p$ that occur as the order of a point $P \in E(K)$ on an elliptic curve $E$ defined over a number field $K$ of degree $d$. We discuss how some plausible conjectures on the sparsity of newforms with certain properties would allow us to deduce a fairly precise result on the asymptotic behavior of $\max S(d)$ as $d$ tends to infinity.

This is joint work with Maarten Derickx.

14h: Adam Morgan (University of Cambridge).

Hasse principle for intersections of two quadrics via Kummer surfaces

I will discuss recent work with Skorobogatov in which we establish the Hasse principle for a broad class of degree 4 del Pezzo surfaces (including all those with irreducible characteristic polynomial), conditional on finiteness of Tate--Shafarevich groups of abelian surfaces. A corollary of this work is that the Hasse principle holds for smooth complete intersections of two quadrics in P^n for n\geq 5, conditional on the same conjecture. This was previously known by work of Wittenberg assuming both finiteness of Tate--Shafarevich groups of elliptic curves and Schinzel's hypothesis (H). 

I will also discuss forthcoming work with Lyczak which, again under the Tate--Shafarevich conjecture, shows that the Brauer--Manin obstruction explains all failures of the Hasse principle for certain degree 4 del Pezzo surfaces about which nothing was known previously.  

15h30: Emiliano Ambrosi (Université de Strasbourg).

Un critère d'Artin et Mumford pour les fibrés en coniques en caractéristique 2

Dans les années 70, Artin et Mumford ont montré un critère d'irrationalité pour un fibré en coniques sur un corps de caractéristique différente de 2 et ils ont construit des exemples sur lesquels il peut être utilisé. 

Dans un travail en cours avec Giuseppe Ancona, on prouve un analogue en caractéristique 2 et on trouve des exemples auxquels s'applique.

13 décembre 2024 (Sorbonne Université)

Konstantinos Kartas (Institut Max-Planck, Bonn).

Azur Đonlagić (Université Paris-Saclay).

Brauer-Manin obstructions on homogeneous spaces of affine algebraic groups over global function fields

Given a family of varieties X over a global field k, one is interested in the sufficiency of the Brauer-Manin obstruction to explain the possible failure of the Hasse principle and weak/strong approximation of adelic points on X. Let G be an affine algebraic group G over k, and our family of interest - the principal homogeneous spaces X of G.

In 1981, Sansuc proved this sufficiency for connected G over a number field k by reduction to the case of a torus and an application of Poitou-Tate duality. Since then, arithmetic duality theorems have proven useful in the study of similar problems. 

In this lecture, we briefly recall the significant generalization by Rosengarten of the Poitou-Tate theory to all commutative affine algebraic groups G over a global field k of any characteristic. Then we explain how this theory allows us to extend the stated Brauer-Manin results to (the principal homogeneous spaces of) all such G, not necessarily smooth or connected, highlighting the difficulties which appear in the case when k is a global function field.

10 janvier 2025 (Sorbonne Université)

Zhizhong Huang (Chinese Academy of Sciences, Beijing).

Raman Parimala (Emory University, Atlanta).