Oberseminar Zahlentheorie und Arithmetische Geometrie

Wintersemester 2020/21

Freitags, 13:00-14:00 Uhr

Online oder Hörsaal F102 im Welfenschloss (Hauptgebäude, Welfengarten 1)

Ulrich Derenthal, Matthias Schütt


Fr 13.11.2020

Julian Lyczak (IST Austria)

Manin's conjecture for symmetric squares of surfaces

One topic in arithmetic geometry is the study of points on a variety over one fixed number field. This talk will be about the study of points over all quadratic extension of the base field simultaneously. For the study of rational points many techniques and conjectures are available. We can also apply these to the study of quadratic points by consider the symmetric square of the variety; any quadratic point on a variety is naturally a rational point on its symmetric square.

During the talk we will count points of bounded height on the symmetric square of some surfaces and compare these results with the results predicted by a class of conjectures first attributed to Manin. Other relevant conjectures we will encounter come from work of Batyrev, Peyre and Tschinkel. I will report on the successes in verifying these conjectures for specific surfaces and failures in trying to do so for a general del Pezzo surface.

This talk is based on joint work with Nils Gubela, and Francesca Balestrieri, Kevin Destagnol, Jennifer Park and Nick Rome.

Fr 27.11.2020Margaret Bilu (IST Austria)

Motivic Euler products and Bertini theorems

The Grothendieck group of varieties over a field k is the quotient of the free abelian group of isomorphism classes of varieties over k by the so-called cut-and-paste relations. It moreover has a ring structure coming from the product of varieties. Many problems in number theory have a natural, more geometric counterpart involving elements of this ring. Thus, Poonen's Bertini theorem over finite fields has a motivic analogue due to Vakil and Wood, which expresses the motivic density of smooth hypersurface sections as the degree goes to infinity in terms of a special value of Kapranov's zeta function. I will report on joint work with Sean Howe, providing a broad generalization of Vakil and Wood's result, which implies in particular a motivic analogue of Poonen's Bertini theorem with Taylor conditions, as well as motivic analogues of many generalizations and variants of Poonen's theorem. A key ingredient for this is a notion of motivic Euler product which allows us to write down candidate motivic probabilities.

Fr 4.12.2020Florian Wilsch (IST Austria)

Equidistribution and freeness on Grassmanians

We associate a tangent lattice to a primitive integer lattice and study its typical shape. This is motivated by Peyre’s program on the freeness of rational points on Fano varieties: A primitive integer lattice can be regarded a point on a Grassmanian, and the shape of its tangent lattice determines this point’s freeness.

The reason behind this interest in freeness is Manin’s conjecture about the number of rational points of bounded height on Fano varieties: This number might be dominated by “bad” points on subvarieties, or more generally, a thin set of “bad“ points that has to be excluded in the count. Peyre proposed to exclude points of low freeness, so that points of high freeness should conform to the asymptotic formula proposed by Manin’s conjecture and its variants. Our analysis verifies this for Grassmanians by proving that there are relatively few points of low freeness.

This is joint work with Tim Browning and Tal Horesh.

Fr 11.12.2020
Fr 18.12.2020
Fr 15.1.2021
Fr 22.1.2021
Fr 29.1.2021