Bath Arithmetic Geometry Seminar

The Bath Arithmetic Geometry Seminar (BAGS) is a fortnightly seminar at the University of Bath on Arithmetic Geometry with a view towards rational points; run by Elyes Boughattas, Julian Demeio, Harry Shaw.

Location: The Wolfson Lecture Theatre, Building 4 West, University of Bath.

Date & Time:  Every other week on Tuesday, 15:15 - 16:15.

To sign up to our mailing list, please email Harry Shaw: hcs50 at bath.ac.uk.

Speaker: Jessica Alessandrì (University of Bath) - 24th September 2024

Title: A local-global problem for divisibility in algebraic groups. 

Abstract: Local-global principles have been widely studied during the last century. In this talk, I will present one of them, the so-called Local-Global Divisibility Problem for commutative algebraic groups. It was first stated in 2001 by Dvornicich and Zannier as a modified version of the Hasse principle for quadratic forms. During the last twenty years, several results have been produced for different algebraic groups. I will present some results in the case of algebraic tori. In particular, I will show a generalization of the Grunwald-Wang Theorem to any algebraic tori with bounded dimension and exhibit a counterexample for the local-global divisibility by any power of an odd prime (joint work with Rocco Chirivì and Laura Paladino). To conclude, I will talk about some work in progress with Laura Paladino in the case of elliptic curves.

Speaker: Jesse Pajwani (University of Bath) - 8th October 2024

Title: 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(Ak), 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 a similar statement holds for X(\A_k). It is a reasonable question to ask whether these obstruction sets also have this property: i.e., is it true that X(A_k)^{obs}=(X_L(\A_L)^{obs})^{Gal(L/k)}? 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 reduce the problem of finding rational points on isotrivial curves over global function fields to the case where our isotrivial curve is constant.

Speaker: Vandita Patel (University of Manchester) - 22nd October 2024

Title: Shifted powers in Lucas-Lehmer sequences

Abstract: The explicit determination of perfect powers in (shifted) non-degenerate, integer, binary linear recurrence sequences has only been achieved in a handful of cases. In this talk, we combine bounds for linear forms in logarithms with results from the modularity of elliptic curves defined over totally real fields to explicitly determine all shifted powers by two in the Fibonacci sequence. A major obstacle that is overcome in this work is that the Hilbert newspace which we are interested in has dimension 6144. We will focus on how this space is computationally handled with respect to the  underlying Diophantine equation. This is joint work with Mike Bennett (UBC) and Samir Siksek (Warwick).

Speaker: Akinari Hoshi (Niigata University) - 5th November 2024

Title: Norm one tori and Hasse norm principle 

Abstract: Let k be a field and T be an algebraic k-torus. In 1969, over a global field k, Voskresenskii proved that there exists an exact sequence $0\to A(T)\to H^1(k,{\rm Pic}\,\overline{X})^\vee\to Sha(T)\to 0$ where A(T) is the kernel of the weak approximation of T, Sha(T) is the Shafarevich-Tate group of T, X is a smooth k-compactification of T, ${\rm Pic}\,\overline{X}$ is the Picard group of $\overline{X}=X\times_k\overline{k}$ and $\vee$ stands for the Pontryagin dual. On the other hand, in 1963, Ono proved that for the norm one torus T=R^{(1)}_{K/k}(G_m) of K/k, Sha(T)=0 if and only if the Hasse norm principle holds for K/k. First, we determine $H^1(k,{\rm Pic}\, \overline{X})$ for algebraic k-tori T up to dimension 5. Second, we determine $H^1(k,{\rm Pic}\, \overline{X})$ for norm one tori $T=R^{(1)}_{K/k}(G_m)$ with $[K:k]\leq 17$. Third, we give a necessary and sufficient condition for the Hasse norm principle for K/k with $[K:k]\leq 15$. We also show that $H^1(k,{\rm Pic}\, \overline{X})=0$ or Z/2Z for $T=R^{(1)}_{K/k}(G_m)$ when the Galois group of the Galois closure of K/k is the Mathieu group M_{11} or the Janko group J_1. As applications of the results, we get the group T(k)/R of R-equivalence classes over a local field k via Colliot-Th\'{e}l\`{e}ne and Sansuc's formula and the Tamagawa number $\tau(T)$ over a number field k via Ono's formula $\tau(T)=|H^1(k,\widehat{T})|/|Sha(T)|$. This is joint work with Kazuki Kanai and Aiichi Yamasaki.  

Speaker: Tim Santens (University of Cambridge) - 3rd December 2024

Title: T.b.c. 

Abstract: T.b.c.