Variétés rationnelles

Le séminaire aura lieu le jeudi 14 novembre au CMLS

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.