Seminar on Selmer Schemes, Spring/Summer 2023

26 June, 2023, 17:30 (UK)

Jennifer Balakrishnan (BU)

Title: Computing Selmer sets

Abstract: The Chabauty--Kim program for studying rational points on hyperbolic curves yields a series of refined Selmer sets at depth n, cut out by n-fold iterated p-adic integrals. We discuss a collection of examples of computations that have been done of these sets in depth 1 and 2, in cases including punctured elliptic curves and smooth projective curves of genus 2 and 3.

19 June, 2023, 17:30 (UK)

Padmavathi Srinivasan (BU)

Title: Towards a unified theory of canonical heights on abelian varieties

We will outline a new construction of canonical p-adic heights on abelian varieties from p-adic adelic metrics, using p-adic Arakelov theory developed by Besser. This construction closely mirrors Zhang's construction of canonical real valued heights from real-valued adelic metrics. We will use this new construction to give direct explanations (avoiding p-adic Hodge theory) of the key properties of height pairings needed for the quadratic Chabauty method for rational points. Time permitting, we will also outline a q-analytic method to compute the local contribution at q to p-adic heights. This is joint work in progress with Amnon Besser and Steffen Mueller.

12 June, 2023, 17:30 (UK)

Minhyong Kim (ICMS, Edinburgh)

Title: Refining Weil's Trichotomy

Abstract: The analogy between function fields and number fields is sometimes extended to a trichotomy that includes meromorphic functions on a compact Riemann surface. It is well-known that this trichotomy is not quite right. It is also well-known how to fix it. In this talk, I will give a discussion of this from the point of view of dimensions, the analogy between primes and knots, and topological quantum field theory.

5 June, 2023, 17:30 (UK)

Netan Dogra (KCL)

Title: How to do a p-descent on a Selmer scheme

Abstract: In this talk I will consider two questions. The first is how to prove finiteness of the sets X(Qp)_n for a given curve X. The second is how to determine the sets X(Qp)_n -- in particular how to solve for the "undetermined coefficients" which have been discussed in some of the other talks in this series. For the first, I will explain how one can "do p-descent for Bloch--Kato Selmer groups", especially when p=2 and the Selmer groups are associated to the wedge square of the Jacobian of a hyperelliptic curve. For the second, I will explain for example that when n=2 (but with no assumption on the Picard number of the Jacobian of X) this has some relation to old questions regarding zero-cycles on surfaces.

29 May, 2023, 17:30 (UK)

Benjamin Moore (Warwick)

Title: The De Rham period map for punctured elliptic curves and the KZB equation

Abstract: For level 2, the de Rham period map associated with the full K_v-prounipotent fundamental group U of a projective or once-punctured projective curve (of genus >1) is well-understood. For higher levels, Beacom provided a very complicated iterative alogrithm which produces the level n de Rham period map from the one at level n-1. However, finding a direct characterisation of the de Rham period map for any infinite dimensional quotient of U remains challenging. We explain how to use the elliptic KZB connection of Levin--Racinet and Luo to solve this problem for punctured elliptic curves with respect to the maximal metabelianisation of U

22 May, 2023, 17:30 (UK)

Martin Lüdtke (Groningen)

Title: Mixed Tate Selmer schemes beyond the polylog quotient

Abstract: Let X be the thrice-punctured line over the ring of S-integers Z_S. The motivic Selmer scheme by definition parametrises G_S-equivariant torsors under the unipotent de Rham fundamental group U of X, where G_S is the mixed Tate Galois group of Z_S. I will show how to define coordinates on the motivic Selmer scheme which make it an affine space. I explain how this can be used for explicit Chabauty--Kim calculations. Previous calculations have been restricted to the polylog quotient of U. By working with the full fundamental group, we find additional Coleman functions on X(Z_p), involving not only polylogarithms Li_k but more generally multiple polylogarithms Li_{k_1,...,k_r}. This is joint work in progress with David Corwin and Ishai Dan-Cohen.

15 May, 2023, 17:30 (UK)

Alex Betts (Harvard)

Title: Chabauty--Kim and the Section Conjecture for locally geometric sections

Abstract: Let X be a smooth projective curve of genus at least 2 over the rational numbers. A natural variant of Grothendieck's Section Conjecture postulates that every section of the fundamental exact sequence for X which everywhere locally comes from a point of X in fact globally comes from a point of X. In this talk, I will show that X satisfies this version of the Section Conjecture if it satisfies Kim's Conjecture for almost all choices of auxiliary prime p, and give the appropriate extension to S-integral points on affine hyperbolic curves. This gives a new ``computational'' strategy for proving instances of this variant of the Section Conjecture, and I will explain how we carry this out for the thrice-punctured line over Z[1/2]. This is joint work with Theresa Kumpitsch and Martin Lüdtke.

8 May, 2023, 17:00 (UK)

David Corwin (Ben-Gurion)

Titile: Explicit Motivic Chabauty-Kim Method