June 30, 2023: TU Wien

Location: "Zeichensaal 3"

Freihaus, tower A, 7th floor

Wiedner Hauptstr. 8-10, 1040 Wien

Time: June 30, 13:00-18:00

Speakers:

Andrew O'Desky (Princeton): "Polynomials with abelian Galois group and integral points on singular toric varieties"

Abstract: How many monic polynomials with bounded integer coefficients have a given Galois group? This can be interpreted as a Diophantine problem on certain quotient varieties, which for abelian Galois groups turn out to be singular toric varieties. In this talk we will discuss how to solve this Diophantine problem in the abelian setting to obtain an asymptotic formula of the form cH^a*log^(b-1)(H)(1+o(1)).

Stephanie Chan (ISTA): "Integral points in families of elliptic curves"

Abstract: Taking a family of elliptic curves and imposing some ordering, we may ask how often a curve has an integral point and how many integral points there are on average. We expect that elliptic curves with any nontrivial integral points are generally very sparse. In certain quadratic and cubic twist families, it is possible to show that almost all curves contain no nontrivial integral points. The proof uses the parameterisation of integral points by integral binary forms and the distribution of Selmer groups. 

Niclas Technau (TU Graz): "Rational points near homogeneous hyper-surfaces."

Abstract: How many rational points are near a compact hyper-surface? We survey the state of the art, explain a standard random model, and indicate (number theoretic) applications. Furthermore, we report on recent joint work with Rajula Srivastava (Uni/MPIM Bonn) on homogeneous hyper-surfaces. Our arguments are rooted in Fourier analysis and, in particular, clarify the role of curvature in the random model.