Oberseminar Zahlentheorie und Arithmetische Geometrie

Fr 13.11.2020

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.

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.

On the distribution of rational points on covers of abelian varieties

In this talk I will discuss results concerning the scarcity of rational points on covers of abelian varieties. In particular, I will show that for every abelian variety A over a number field K, every ramified, irreducible cover X of A, and every subgroup Omega of A(K) that is Zariski-dense in A, there is a finite-index coset of points of Omega that do not lift to K-points of X. I will also show that, under an additional (necessary) assumption, the fibers of such a cover over each point in a finite-index coset of Omega are even irreducible over K. This confirms a conjecture of Corvaja and Zannier concerning their "generalized Hilbert property" for rational points in the case of abelian varieties.

This is joint work with P. Corvaja, J. Demeio, A. Javanpeykar, and U. Zannier.

Fr 8.1.2021 (10:00)

Campana points and powerful values of norm forms

The theory of Campana points is of growing interest in arithmetic geometry due to its ability to interpolate between the notions of rational and integral points. Further, it naturally lends itself to studying “arithmetically interesting” solutions of equations. In this talk, I will introduce Campana points and explain the key ideas and principles behind recent results on asymptotics for Campana points of bounded height, providing evidence for a Manin-type conjecture proposed in work of Pieropan, Smeets, Tanimoto and Várilly-Alvarado. I will also indicate how these results give rise to an asymptotic formula for powerful (e.g. square-full) values of norm forms.

Rational points on del Pezzo surfaces of degree 4

In this talk I shall explain what is the frequency of failures of certain local-to-global principles in a family of del Pezzo surfaces of degree four. This is addressed in terms of the Brauer group of such surfaces and more precisely by describing its generators explicitly and incorporating the information about them in the Brauer-Manin obstruction. In this way we obtain sharp upper and lower bounds for the number of surfaces with a Brauer group of fixed order as well as bounds for the number of Hasse and weak approximation failures in the family. The talk is based on a joint work with Cecília Salgado.

Local-global principles for norms

Given an extension L/K of number fields, we say that the Hasse norm principle (HNP) holds if every non-zero element of K which is a norm everywhere locally is in fact a global norm from L. If L/K is cyclic, the original Hasse norm theorem states that the HNP holds. More generally, there is a cohomological description (due to Tate) of the obstruction to the HNP for Galois extensions.

In this talk, I will present work developing explicit methods to study this principle for non-Galois extensions. As a key application, I will describe how these methods can be used to characterize the HNP for extensions whose normal closure has Galois group A_n or S_n. I will additionally discuss some recent generalizations of these methods to study the Hasse principle and weak approximation for products of norms as well as consequences in the statistics of these local-global principles.