Monday 16th December 2024.
S-3.20, Floor Minus 3, Strand Building, Strand Campus, King’s College London.
Schedule
10:55 – 11:00 welcome
11:00 – 11:30 Lazar Radicevic
11:30 – 12:05 coffee (S-3.18)
12:05 – 12:35 Harvey Yau
12:35 – 13:05 Jessica Alessandrì
13:05 – 14:30 lunch
14:30 – 15:00 Giorgio Navone
15:00 – 15:30 Jesse Pajwani
15:30 – 16:15 coffee (S-3.18)
16:15 – 16:45 Julie Tavernier
16:45 – 17:15 Harmeet Singh
18:00 dinner
Speakers
Jessica Alessandrì: https://sites.google.com/view/jessicaalessandri/home
Giorgio Navone: https://sites.google.com/view/giorgio-navone/home
Jesse Pajwani: https://sites.google.com/view/jessepajwani/Jesse-Pajwani
Lazar Radicevic: https://lazaradicevic.github.io
Harmeet Singh: https://sites.google.com/view/harmeetsingh/
Julie Tavernier
Harvey Yau: https://tkhy2.user.srcf.net
Abstracts
Jessica Alessandrì: A local-global problem for divisibility in algebraic groups
Abstract: In this talk I will present a Hasse principle on divisibility of points in algebraic groups, introduced by Dvornicich and Zannier in 2001, motivated by a particular case of the Hasse principle on quadratic forms and by the Grunwald-Wang Theorem. During the last twenty years, several results have been produced for different algebraic groups. I will give an overview on some recent developments, in particular for the case of algebraic tori and for elliptic curves. This is based on a joint work with Chirivì and Paladino.
Giorgio Navone: Transcendental Brauer groups of generalized Kummer surfaces
Abstract: The transcendental Brauer group of a variety is a troublesome cohomological invariant that lacks a general strategy of approach. In this talk, we will present ongoing work computing the transcendental Brauer group for a family of K3 surfaces, constructed from a planar cubic curve in a similar fashion to Kummer surfaces. If time permits, we’ll illustrate the difficulty in dealing with the Hasse principle for these varieties.
Jesse Pajwani: Galois invariants of obstruction sets
Abstract: Let k be a global field and let X be a variety over k. We are often interested in how the set of rational points X(k) sits inside the set of adelic points X(A_k), and we often develop obstruction sets X(A_k)^{obs} such that X(k)⊆X(A_k)^{obs}⊆X(A_k). Examples of these obstruction sets include the topological closure of the k points, the Brauer-Manin obstruction, or the finite étale descent obstruction. The sets X(k) and X(A_k) have an additional structure coming from Galois theory: if L/k is a finite Galois extension, then X(k)=X_L(L)^{Gal(L/k)}, and similarly for X(A_k). It is a reasonable question to ask whether these obstruction sets also have this property. In this talk, I'll give a survey of joint work with Creutz and Voloch that shows that for many natural obstructions, the above principle holds when X is a subvariety of an abelian variety. I'll also show how we can construct varieties such that the above principle fails for these natural obstructions. Finally, I'll discuss how the above work allows us to obtain new rational points results.
Lazar Radicevic: 3-descent on genus 2 Jacobians using visibility
Abstract: We show how to explicitly compute equations for everywhere locally soluble 3-coverings of Jacobians of genus 2 curves with a rational Weierstrass point, using the notion of visibility introduced by Cremona and Mazur. These 3-coverings are abelian surface torsors, embedded in the projective space P^8 as degree 18 surfaces. They have points over every p-adic completion of Q, but no rational points, and so are counterexamples to the Hasse principle and represent non-trivial elements of the Tate-Shafarevich group. Joint work in progress with Tom Fisher.
Harmeet Singh: The Hasse norm principle for wreath product extensions
Abstract (co-authored by CHATGPT): For an extension K/k of number fields, we say that the Hasse norm principle (HNP) holds if an element of k is a global norm whenever it is a norm everywhere locally. A natural question arises: for a fixed base field k and degree d, what proportion of degree-d extensions of k satisfy HNP? Addressing this question leads naturally to the study of so-called wreath product extensions. In this talk, I will discuss ongoing work (joint with Jiuya Wang) on determining when HNP holds for these wreath product extensions and what this tells us about proportions of extensions with HNP.
Julie Tavernier: Number fields with restricted ramification and rational points on stacks
Abstract: A conjecture by Malle gives a prediction for the number of number fields of bounded discriminant. In this talk I will give an asymptotic formula for the number of abelian number fields of bounded height with restricted ramification and provide an explicit formula for the leading constant in terms of sums of Euler products. I will then discuss how one can view this problem as one on counting rational points on the stack BG, and that the existence of such number fields is controlled by a Brauer-Manin obstruction.
Harvey Yau: Descent on elliptic surfaces and Brauer-Manin obstruction
Abstract: In this talk I will discuss how to calculate elements of the Brauer group explicitly for elliptic surfaces. The method used for this shares many similarities with that of descent on an elliptic curve defined over a number field, but also brings its own unique aspects. I will also show applications of this to Brauer-Manin obstruction of weak approximation on elliptic curves.