2020

Intercity Number Theory Seminar

11:15–12:15

Adelina Mânzăţeanu UL, Rational curves on cubic hypersurfaces over 𝔽_q

Using a version of the Hardy–Littlewood circle method over 𝔽_q(t), one can count 𝔽_q(t)-points of bounded degree on a smooth cubic hypersurface X ⊂ ℙⁿ⁻¹ over 𝔽_q. Moreover, there is a correspondence between the number of 𝔽_q(t)-points of bounded height and the number of 𝔽_q-points on the moduli space which parametrises the rational maps of degree d on X. In this talk I will give an asymptotic formula for the number of 𝔽_q-rational curves on X passing through two fixed points, one of which does not belong to the Hessian, for n ≥ 10, and q and d large enough. Further, I will explain how to deduce results regarding the geometry of the space of such curves.

13:30–14:30

Adam Morgan MPIM Bonn, Parity of Selmer ranks in quadratic twist families

We study the parity of 2-Selmer ranks in the family of quadratic twists of a fixed principally polarised abelian variety over a number field. Specifically, we prove results about the proportion of twists having odd (resp. even) 2-Selmer rank. This generalises work of Klagsbrun–Mazur–Rubin for elliptic curves and Yu for Jacobians of hyperelliptic curves. Several differences in the statistics arise due to the possibility that the Shafarevich–Tate group (if finite) may have order twice a square.

14:45–15:45

Rosa Winter UL, Density of rational points on a family of del Pezzo surfaces of degree 1

Let k be a number field and X an algebraic variety over k. We want to study the set of k-rational points X(k). For example, is X(k) empty? And if not, is it dense with respect to the Zariski topology? Del Pezzo surfaces are surfaces that are classified by their degree d (for d ≥ 3, these are the surfaces of degree d in ℙ^d). For all del Pezzo surfaces of degree ≥ 2 over k, we know that the set of k-rational points is dense provided that the surface has a k-rational point (that lies outside a specific subset of the surface for degree 2). But for del Pezzo surfaces of degree 1 over k, even though we know that they always contain a k-rational point, we do not know whether the set of k-rational points is dense. In this talk I will focus on a result that is joint work with Julie Desjardins, in which we prove that for a specific family of del Pezzo surfaces of degree 1 over k, under a mild condition, the k-rational points are dense with respect to the Zariski topology. I will compare this to previous results.

16:00–17:00

Peter Koymans MPIM Bonn, On Chowla’s conjecture over function fields

Let χ be a quadratic Dirichlet character. Then Chowla’s conjecture states that L(1/2, χ) is non-zero. Over function fields this conjecture has recently been disproved by Wanlin Li. The main result of this talk is that for many values of s we have that L(s, χ) is not zero for 100% of the characters χ. This is work in progress with Carlo Pagano and Mark Shusterman.

Belgian-Dutch Algebraic Geometry seminar

Intercity Number Theory Seminar

11:00–12:00

Jessica Fintzen Michigan / Cambridge, Representations of p-adic groups

The Langlands program is a far-reaching collection of conjectures that relate different areas of mathematics including number theory and representation theory. A fundamental problem on the representation theory side of the Langlands program is the construction of all (irreducible, smooth, complex) representations of p-adic groups. I will provide an overview of our understanding of the representations of p-adic groups, with an emphasis on recent progress. I will also briefly discuss applications to other areas, e.g. to automorphic forms and the global Langlands program.

13:30–14:30

Per Salberger Chalmers, Equal sums of three kth powers

I will present new uniform estimates for the number of rational points of bounded height on non-singular threefolds in ℙ⁴ . These are obtained by means of a global version of Heath-Brown's p-adic determinant method and recent geometric results on covering gonality. As a corollary we obtain new estimates for equal sums of three powers, which are much stronger than the previous estimates of Hua, Browning/Heath-Brown and myself.

14:45–15:45

Sean Prendiville Lancaster, Counting monochromatic solutions to diagonal Diophantine equations

The Hardy–Littlewood circle method is a cornerstone of analytic number theory, and is particularly adept at counting the number of solutions to a system of diagonal Diophantine equations in sufficiently many variables. The method has been adapted to the situation in which variables are restricted to sets of arithmetic interest, such as the primes or smooth numbers. However, what if the variables are restricted to a set for which we have little arithmetic information, beyond the fact that it has positive density? We shall discuss recent developments illustrating how to get the circle method to work in this context, with consequences in arithmetic combinatorics and Ramsey theory. This includes results joint with T. Browning, and S. Chow and S. Lindqvist.

16:00–17:00

Quoc P. Ho IST Austria, Homological densities and stability of generalized configuration spaces

Using factorization homology, we develop a uniform and conceptual approach for treating homological stability, homological densities, and arithmetic densities of generalized configuration spaces in algebraic geometry. This categorifies and generalizes the coincidences appearing in the work of Farb–Wolfson–Wood, and in fact, provides a conceptual understanding of these coincidences. Our approach hinges on the study of homological stability of cohomological Chevalley complexes, which is of independent interest.

Intercity Number Theory Seminar

Belgian-Dutch Algebraic Geometry seminar

Intercity Number Theory Seminar

Intercity Number Theory Seminar / Getaltheorie in het vlakke land