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, Julie Tavernier.
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.
Date: 3rd June 2025
Speaker: Céline Maistret (University of Bristol)
Title: Proving the parity conjecture for abelian varieties: a local discrepancy problem.
Abstract: The Birch and Swinnerton-Dyer conjecture (BSD) famously links the rank of abelian varieties to the order of vanishing at s=1 of their L-function. A weaker version of this conjecture is the parity conjecture, which asserts that the parity of the rank of abelian varieties is given by an algebraic invariant called their root number. In this talk, I will recall how to derive the parity conjecture from BSD and, using elliptic curves as examples, I will present a general strategy to prove it.
One step of this strategy will be my main focus: identifying a local discrepancy between local invariants of the variety. For example, in the case of elliptic curves, this local discrepancy takes the form of a product of two Hilbert symbols. Unfortunately, we currently have no conceptual interpretation for this local discrepancy and this proves to be a bottleneck to generalising this approach to higher dimensional abelian varieties.
Date: 20th May 2025
Speaker: Béranger Seguin (Padderborn University)
Title: Counting wildly ramified extensions of function fields
Abstract: The asymptotic distribution of field extensions (often counted by discriminant) is an active topic that aims to make the predictions of inverse Galois theory quantitative.
Over number fields, precise conjectures and partial results give a clear idea of the situation, and a similar picture emerges for tamely ramified extensions of function fields.
In contrast, the wildly ramified case remains a terra incognita: the case of abelian extensions was only recently (partially) solved using class field theory.
In this talk, we fix a prime p>2, and we focus on p-groups G of nilpotency class 2 (the "most abelian" among non-abelian p-groups).
We will explain how to parametrize G-extensions of fields of characteristic p.
Work by Abrashkin describes the ramification filtration of these extensions, and reduces the counting problem (over local fields) to a "typical" problem of arithmetic geometry: counting solutions to polynomial equations over finite fields.
We present several instances where this local counting can be carried out.
If time permits, we will also discuss results over global function fields, which follow from the local case via a local-global principle.
This work is a joint collaboration with Fabian Gundlach.
Date: 6th May 2025
Speaker: Loïs Faisant (ISTA)
Title: Counting rational curves with prescribed tangency conditions: a motivic analogue via universal torsors
Abstract: Given a smooth projective and geometrically irreducible curve C and a Mori Dream Space X, we present a general parametrisation of morphisms from C to X which allows us to express the Grothendieck motive of Hom (C,X) as a motivic function defined on some power of the scheme of effective divisors of C, generalising previous works of Bourqui. Such a parametrisation should be understood as lifting our morphisms to the universal torsor of X.
As an application, we prove a motivic analogue of a variant of Manin’s conjecture for Campana curves on smooth projective split toric varieties.
(arXiv:2502.11704)
Date: 29nd April 2025
Speaker: Robin Bartlett (University of Glasgow)
Title: Moduli spaces of p-adic Galois representations, and their connections to representation theory
Abstract: In the first half of the talk I will motivate the study of moduli spaces of p-adic Galois representations (of fields like Qp) by explaining how sufficient knowledge of their geometry can be used to prove difficult results in number theory. A byproduct of these ideas is that these moduli spaces are also expected to support interesting connections to the representation theory of GLn. While these connections are in general very mysterious, in some specific cases the relevant moduli spaces can be described concretely, and their geometry directly related to representation theory. In the second part of the talk I will discuss results in this direction, which indicate connections to both known and (as far as I am aware) new phenomena in geometric representation theory. No background in p-adic Hodge theory will be assumed.
Date: 22nd April 2025
Speaker: Sam Frengley (University of Bristol)
Title: Galois groups of abelian varieties over finite fields and exceptional Tate classes
Abstract: This talk is based on joint work with Santiago Arango Piñeros and Sameera Vemulapalli.
Let A/F_q be a simple abelian variety and consider the free abelian group Z^r(A) of cycles on A of codimension r. The Tate conjecture predicts that (the Galois invariants of) appropriate cohomology groups H^2r(A,Q_l(r)) are spanned by Z^r(A) (compare with the Hodge conjecture). In the 1960s Tate proved that this conjecture holds when r=1 (i.e., for divisors on A). Since then a fruitful strategy for proving the Tate conjecture for specific abelian varieties has been to leverage Tate's result and study when intersections of divisors span H^2r(A,Q_l(r)) -- this has been employed by Tankeev, Zarhin, Lenstra--Zarhin, Dupuy--Kedlaya--Zureick-Brown. I will discuss a refinement of this approach. In particular we will see the role of the interactions between the eigenvalues of the Frobenius endomorphism acting on A, and how to package this as a certain representation of a Galois group over Q (the Galois group of the minimal polynomial of Frobenius). Time permitting, we will discuss some relevant inverse Galois problems.
Date: 25th March 2025
Speaker: Azur Đonlagić (Université Paris-Saclay)
Title: Brauer-Manin obstructions on homogeneous spaces of commutative affine algebraic groups over global function fields
Abstract: Given a family of varieties X over a global field k, one is interested in the sufficiency of the Brauer-Manin obstruction to explain the possible failure of the Hasse principle and weak/strong approximation of adelic points on X. Let G be an affine algebraic group G over k, and our family of interest - the principal homogeneous spaces X of G.
In 1981, Sansuc proved this sufficiency for connected G over a number field k by reduction to the case of a torus and an application of Poitou-Tate duality. Since then, arithmetic duality theorems have also proven useful in the study of many similar problems.
In this lecture, we briefly recall the significant generalization by Rosengarten of the Poitou-Tate theory to all commutative affine algebraic groups G over a global field k of any characteristic. Then we explain how this theory allows us to extend the stated Brauer-Manin results to (the principal homogeneous spaces of) all such G, not necessarily smooth or connected, highlighting the difficulties which appear in the case when k is a global function field.
This talk is based on the speaker's recent preprint, available on arXiv.
Date: 25th February 2025
Speaker: Alec Shute (University of Bristol)
Title: Local solubility of generalised Fermat equations
Abstract: I will discuss recent work with Peter Koymans, Ross Paterson and Tim Santens in which we determine, for every positive integer n, an asymptotic formula for the number of integer triples (a,b,c) of bounded height such that the equation ax^n + by^n + cz^n = 0 is everywhere locally soluble. Our result is compatible with conjectures of Loughran Smeets and Loughran, Rome, Sofos on everywhere local solubility in fibrations.
Date: 11th February 2025
Speaker: Boaz Moerman (University of Utrecht)
Title: Generalized Campana points and adelic approximation
Abstract: In recent years there has been considerable interest into Campana points, which provide a geometric framework for studying questions involving powerful numbers. We will considerably generalize these points to “M-points”, to also be able to study squarefree numbers, powers and more in a geometric setting. For these M-points, we study when these points are dense in the corresponding space of adelic points. We find a simple combinatorial criterion which characterizes when this analogue of strong approximation is satisfied on a toric variety. This generalizes and strengths classical results on strong approximation and generalizes recent work on Campana weak approximation by Nakahara and Streeter.
Date: 28th January 2025
Speaker: Haowen Zhang (University of Leiden)
Title: Description of the strong approximation locus using Brauer-Manin obstruction for homogeneous spaces with commutative stabilizers
Abstract: For a homogeneous space $X$ over a number field $k$, the Brauer-Manin obstruction has been used to study strong approximation for $X$ away from a finite set $S$ of places, and known results state that $X(k)$ is dense in the omitting-$S$ projection of the Brauer-Manin set $\pr_S(X(\A_k)^{\br})$, under certain assumptions. In order to completely understand the closure of $X(k)$ in the set of $S$-adelic points $X(\A_k^S)$, we ask (i) if $\pr_S(X(\A_k)^{\br})$ is closed in $X(\A_k^S)$; (ii) if $X(k)$ is dense in the closed subset of $X(\A_k^S)$ cut out by elements in $\br X$ which induce zero evaluation maps at all the places in $S$. We also ask these questions considering only the algebraic Brauer group. We give answers to such questions for homogeneous spaces $X$ under semisimple simply connected groups with commutative stabilizers.