# The Advent of Rational Points

Monday 12th December 2022.

K-1.56, King’s Building, Strand Campus, King’s College London.

Schedule

10:25 -10:30 welcome

10:30 – 11:00 Sebastian Monnet

11:00 – 11:30 Rosa Winter

11:30 – 12:05 coffee (maths common room S5.21)

12:05 – 12:35 Jesse Pajwani

12:35 – 13:05 Mohamed Alaa Tawfik

13:05 – 14:30 lunch (Franco Manca Aldwych)

14:30 – 15:00 Tim Santens

15:00 – 15:30 Judith Ortmann

15:30 – 16:15 coffee (maths common room S5.21)

16:15 – 16:45 Jakob Glas

16:45 – 17:15 Domenico Valloni

18:00 dinner (Sagar Covent Garden)

Abstracts

Mohamed Alaa Tawfik: Brauer-Manin obstructions on Kummer surfaces of products of CM elliptic curves

Abstract: We discuss Brauer-Manin obstructions on Kummer surfaces of products of certain CM elliptic curves. We start by putting necessary and sufficient conditions on these surfaces to get a non-trivial transcendental Brauer group, then we find a generator of this group. Further, using a theorem by Harpaz and Skorobogatov, we show that a non-trivial element of order 5 of the transcendental Brauer group always gives rise to Brauer-Manin obstruction to weak approximation on these surfaces. Finally, we show that for most cases there is no obstruction coming from the algebraic part.

Jakob Glas: Rational points on del Pezzo surfaces of low degree

Manin’s conjecture gives a prediction for the number of rational points of bounded height on del Pezzo surfaces. However, the conjecture is not known to hold for any single del Pezzo surface of degree at most 3. In this talk I will report on joint work with Leonhard Hochfilzer on obtaining upper bounds for the associated counting function for del Pezzo surfaces of degree at most 3 over global fields of positive characteristic.

Sebastian Monnet: S4-quartics with prescribed norms

Abstract: Let $K$ be a number field with $\mathbb{Q}$-basis $\{e_1, ..., e_n\}$, and let $\alpha$ be a rational number. It is natural to ask whether the “norm equation” $N_{K/\mathbb{Q}}(x_1e_1 + ... + x_ne_n) = \alpha$ has rational solutions. Since the answer depends only on $K$, we may ask how often this norm equation has rational solutions as we vary $K$. The case of abelian number fields was solved by Frei-Loughran-Newton, and in this talk we present one of the simplest non-abelian cases: $S_4$-quartics.

Judith Ortmann: Counting conics over a global function field that have a rational point

Abstract: In this talk we deal with a family of conics defined over projective space over the global function field $\mathbb{F}_2(t)$. We are interested in the number of conics in this family that do have a rational point. The first step is to study which conics in the family do have a rational point. To finally count these conics, the basic approach is to compute the analytic height zeta function by using Poisson summation formula and computing the local Fourier transforms, and then use a Tauberian theorem.

Jesse Pajwani: Arithmetic enumerative geometry the Yau-Zaslow Formula

Abstract: Arithmetic enumerative geometry is a new area, in which we take classical results in enumerative geometry, and refine them to also include arithmetic information. In this talk, I’ll give an introduction to the field of arithmetic enumerative geometry, and eventually build up to stating an arithmetic refinement of the Yau-Zaslow formula for counting rational curves on K3 surfaces. This talk is on joint work with Ambrus Pál.

Tim Santens: Integral and rational points on stacks

Abstract: A stack is a type of geometric object which was first used in the study of moduli problems. I will begin by giving an introduction to stacks, in particular focusing on their integral and rational points. I will then explain how natural arithmetic questions can be interpreted as questions about the integral and rational points of certain stacks and on some recent work of myself and others on how this interpretation can be helpful in answering the original arithmetic problem.

Domenico Valloni: Reduction modulo p of the Noether’s problem

Abstract : Let k be any field and V be a linear and faithful representation of a p-group G. The Noether’s problem asks whether V/G is a (stably) rational variety over k.

If p is equal to char(k), it is known that V/G is always rational; on the other hand, Saltman constructed p-groups G for which V/G is not stably rational over C. We ask what happens over a dvr R of mixed characteristic (0,p):

Is there a smooth projective scheme over R whose special fibre is stably rational and generic fibre is stably birational to V/G?

We show that such model never exists for all the examples found by Saltman and Bogomolov. Incidentally, our result shows that the Artin-Mumford invariant is an obstruction to the existence of such model in general.

This is joint work with Emiliano Ambrosi from Strasbourg.

Rosa Winter: Weak weak approximation for del Pezzo surfaces of degree 2

Abstract: I will talk about rational points on del Pezzo surfaces of low degree. After giving an overview of different abundance notions and what is known so far, I will focus on work in progress joint with Julian Demeio and Sam Streeter on weak weak approximation for del Pezzo surfaces of degree 2.