Winter 2025 Schedule
18/02/25 1pm K4.31: Elvira Lupoian
Computing torsion points on Jacobians of Curves
Points on the Jacobian of a curve can easily be constructed when points on the curve are known. We may ask whether one could compute points on the Jacobian without prior knowledge of any points on the curve. In this talk we discuss a method for computing the group of 3-torsion points on the Jacobian of a genus 3 curve. We use elementary geometry to derive a system of equations whose equations parametrise 3-torsion points and use complex analysis and lattice reduction to find precise expressions for the solutions. Time permitting, I will discuss an application of this to computing the local conductor exponent at 2.
25/02/25: *11am* K2.41 Sudip Pandit
Why Arithmetic Jet Spaces?
The theory of arithmetic jet spaces is rooted in δ-geometry, which has emerged as an elegant and powerful framework in recent advances in p-adic geometry. In this talk, I will provide an overview of arithmetic jet spaces and explore their applications in Diophantine geometry and p-adic Hodge theory. Along the way, I will also present a brief survey of key developments in this area.
04/03/25: 1pm S-1.08 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.
11/03/25: 1pm S-1.08 Zerui Tan
Title: Gauss Manin Connection and p-adic Differential Equations inNumber Theory
Abstract: In this talk, we’ll unravel the origins of the Gauss Manin connection and explore how it helps us tackle families of integrals: a quest to uncover p-adic differential equations linked to families of Coleman integrals. Some dark technical lemmas will also be needed to obtain local bounds and beat the number of zeros of these analytic functions, which gives us arithmetic treasures. You will see a lot of numbers at the end of the talk, definitely number theory!
18/03/25: 1pm K0.19 Giorgio Navone
Title: Quadratic points on del Pezzo surfaces of degree 4
Abstract: This talk will be an overview of the problem of determining the existence of quadratic points on the intersection of two quadrics in five variables. We'll explain the always affirmative answer in the p-adic case, a negative answer in the rational case and finally, time permitting, will discuss ongoing work on C_2 fields.
Autumn 2023 Schedule
10/10/23: Harmeet Singh
Abelian birational sections
24/10/23: Lee Berry
Explicit descent for superelliptic curves
7/11/23: Aashraya Jha
Integral points on elliptic curves over number fields
21/11/23: Alex Best
Generalized Fermat equations
28/11/23: Corijn Rudrum
Selmer group Chabauty
05/12/23: Steffen Müller
p-adic adelic metrics
12/12/23: Izzy Rendell
Quadratic Chabauty for modular curves