# WORKing Seminar

The WORKing seminar is a seminar series on Diophantine geometry jointly organised with Christopher Daw, Tiago Fonseca and Martin Orr . From 2019-2020, the seminar was funded by an LMS Scheme 3 grant.

The next meeting is on Friday May 7th. The speakers are Nirvana Coppola (VU Amsterdam), Adela Gherga (Warwick) and Jef Laga (Cambridge).

Schedule

11-12 Adela Gherga: On the sum of two S-units being a square

1-2 Jef Laga: Rational points and Selmer groups of genus 3 curves

2:30-3:30 Nirvana Coppola: Wild Galois representations of hyperelliptic curves

Abstracts:

On the sum of two S-units being a square

Let p_1, . . . , p_s be a set of distinct rational primes and let S be the set of positive rational integers which have no prime divisors different from the p_i. A rational number is called an S-unit if its absolute value is a quotient of elements of S. Consider the Diophantine equation x + y = z^2 in x, y, S-units, and z ∈ Q. To resolve such an equation for any given set of primes, there exists a practical method of de Weger using bounds for linear forms in p-adic logarithms and various reduction techniques. In this talk, we describe the implementation of this method and discuss the key steps and bottlenecks of this algorithm.

Rational points and Selmer groups of genus 3 curves

Manjul Bhargava and Arul Shankar have determined the average size of the n-Selmer group of the family of all elliptic curves over Q ordered by height, for n at most 5. They used this to show that the average rank of elliptic curves is less than one. In this talk we will consider a family of nonhyperelliptic genus 3 curves, and bound the average size of the 2-Selmer group of their Jacobians. This implies that a majority of curves in this family have relatively few rational points. We also consider a family of abelian surfaces which are not principally polarized and obtain similar results.

Wild Galois representations of hyperelliptic curves

In this talk we will investigate the Galois action on a certain family of hyperelliptic curves defined over local fields. In particular we will look at curves with potentially good reduction, which acquire good reduction over a wildly ramified "large" extension. We will first clarify what large means and then show how to determine the Galois representation in the case considered.

The second meeting of the academic year was held on Thursday February 25th. The speakers were Ekin Özman (Boğaziçi University ) and Francesca Balestrieri (American University of Paris).

Schedule

11-12 Ekin Özman: Quadratic points on modular curves and Fermat-type equations

3-4 Francesca Balestrieri: Strong approximation for homogeneous spaces of linear algebraic groups

Abstracts:

Quadratic points on modular curves and Fermat-type equations

Understanding solutions of Diophantine equations over rationals or more generally over any number field is one of the main problems of number theory. By the help of the modular techniques used in the proof of Fermat’s last theorem by Wiles and its generalizations, it is possible to solve other Diophantine equations too. Understanding quadratic points on the classical modular curve play a central role in this approach. It is also possible to study the solutions of Fermat type equations over number fields asymptotically. In this talk, I will mention some recent results about these notions.

Strong approximation for homogeneous spaces of linear algebraic groups

Building on work by Yang Cao, we show that any homogeneous space of the form G/H with G a connected linear algebraic group over a number field k satisfies strong approximation off the infinite places with étale-Brauer obstruction, under some natural compactness assumptions when k is totally real. We also prove more refined strong approximation results for homogeneous spaces of the form G/H with G semisimple simply connected and H finite, using the theory of torsors and descent. (This latter result is somewhat related to the Inverse Galois Problem.)

The first meeting of 2020-21 was on Monday December 14th. The speakers were Ziyang Gao (Jussieu), Damaris Schindler (Goettingen) and Efthymios Sofos (Glasgow).

Schedule

10-11 Damaris Schindler: On the distribution of Campana points on toric varieties

11.30-12.30 Efthymios Sofos: Möbius randomness law and rational points on surfaces

13.30-14.30 Ziyang Gao : Bounding the number of rational points on curves

Abstracts:

On the distribution of Campana points on toric varieties:

In this talk we discuss joint work with Marta Pieropan on the distribution of Campana points on toric varieties. We discuss how this problem leads us to studying a generalised version of the hyperbola method, which had first been developed by Blomer and Bruedern. We show how duality in linear programming is used to interpret the counting result in the context of a general conjecture of Pieropan-Smeets-Tanimoto-Varilly-Alvarado.

Möbius randomness law and rational points on surfaces

In recent work with Alexei Skorobogatov https://arxiv.org/abs/2005.02998 we prove Schinzel's Hypothesis for almost all polynomials. The hypothesis states that a collection of integer polynomials simultaneously represents primes, standard to necessary obvious assumptions. It includes the twin problem but the only known case is a single polynomial of degree 1. I will focus on the implications of this result to rational points on random Châtelet surfaces. These surfaces are central in the Brauer-Manin theory in arithmetic geometry, however, their arithmetic is poorly understood in the majority of cases.

Bounding the number of rational points on curves

Mazur conjectured, after Faltings’s proof of the Mordell conjecture, that the number of rational points on a curve of genus g at least 2 defined over a number field of degree d is bounded in terms of g, d and the Mordell-Weil rank. In particular the height of the curve is not involved. In this talk I will explain how to prove this conjecture and some generalizations. I will focus on how functional transcendence and unlikely intersections are applied in the proof. If time permits, I will talk about how the dependence on d can be furthermore removed if we moreover assume the relative Bogomolov conjecture. This is joint work with Vesselin Dimitrov and Philipp Habegger.

From August to September 2020, together with the ERLASS arithmetic statistics seminar and the Northern number theory seminar, we hosted eight online lectures on number theory by early career researchers. These were intended to be suitable for beginning PhD students. The speakers were Richard Hatton, Beth Romano and Tiago Fonseca.

Richard Hatton: Heegner points and self-points on elliptic curves

Beth Romano: An introduction to Vinberg theory and related arithmetic invariant theory

Tiago Fonseca: A crash course on modular forms and cohomology

The videos of their lectures may be found here:

https://www.youtube.com/playlist?list=PLsDn5JyJXoYIc2ooNb8tTDtdwTbJQUK-g

Useful references and details for Beth's lectures may be found here:

https://sites.google.com/site/bethromano/teaching/vinberg

Useful references and details for Tiago's lectures may be found here:

http://people.maths.ox.ac.uk/jardimdafons/row.html

Slide for Richard's lectures may be found here and here.

Abstracts:

Richard Hatton -- Heegner points and self-points on elliptic curves

In the arithmetic of elliptic curves, we are interested in the construction of points on an elliptic curve. In particular, it has been shown that we are able to bound certain Selmer groups using modular points, specifically the use of Heegner points by Kolyvagin and self points by Wuthrich. We will define these points and will show how they can be used to create the bounds and its generalisations.

Beth Romano -- An introduction to Vinberg theory and related arithmetic invariant theory

In recent years, Vinberg theory of graded Lie algebras has become relevant in many areas of number theory, from arithmetic statistics (eg in the work of Romano--Thorne) to the local Langlands correspondence (eg in the work of Reeder--Yu). These lectures will provide the algebraic background for number theory students to engage with research involving graded Lie algebras. We'll start by discussing some of the relevant aspects of the invariant theory of Lie algebras, including the Chevalley restriction theorem and the pioneering work of Kostant on invariant rings. We'll then define graded Lie algebras and look at the graded analogues of these theorems, based on work of Vinberg. Time permitting, we'll look at Slodowy slices and applications to families of algebraic curves. These lectures should give number theory students sufficient background to read, for example, Thorne's paper ``Vinberg's representations and arithmetic invariant theory" and other related papers. But the lectures will also be a useful introduction to some beautiful aspects of Lie theory for students in algebra and representation theory. I'll assume students have some knowledge of Lie algebras, but I will review relevant background and provide examples throughout the lectures.

Tiago Fonseca -- A crash course on modular forms and cohomology

This is a geometrically flavoured introduction to the theory of modular forms. We will start with a standard introduction to some basic analytic aspects concerning modular forms and to their interpretation as sections of line bundles on modular curves.

Then, our main goal will be to explain how one can attach certain 2-dimensional cohomology groups to Hecke eigenforms. In this course, we will only deal with algebraic de Rham and Betti cohomology, but this can also serve to build geometric intuition on the l-adic setting, which gives rise to the famous l-adic representations attached to modular forms.

We will finish with a discussion on the Eichler-Shimura isomorphism, periods of modular forms, and, depending on time, Manin's theorem on the critical values of L-functions of modular forms.

From January to June we held three one day events (one in Oxford, the other two remotely).

**Monday the 8th of June.**

**Schedule:**

10:00 - 11:00 Rachel Newton (Reading) Title: Explicit uniform bounds for Brauer groups of singular K3 surfaces

11:30 - 12:30 Gregorio Baldi (UCL) Title: Special subvarieties of non arithmetic ball quotients

13:30 - 14:30 Beth Romano (Oxford) Title: Arithmetic statistics via graded Lie algebras

**Abstracts:**

Title: Explicit uniform bounds for Brauer groups of singular K3 surfaces

Abstract: Várilly-Alvarado has conjectured that Brauer groups (modulo constants) of K3 surfaces over number fields are bounded by a number that only depends on degree of the field and the isomorphism class of theNéron-Severi lattice. Orr and Skorobogatov have proved this conjecture for K3 surfaces of CM type, showing the existence of a bound that only depends on the degree of the number field. I will present joint work with Francesca Balestrieri and Alexis Johnson in which we re-prove Várilly-Alvarado's conjecture for singular K3 surfaces, this time with an explicit bound. When combined with results of Kresch--Tschinkel and Poonen--Testa--van Luijk, this shows that the Brauer--Manin sets for these varieties are effectively computable.

Title: Special subvarieties of non arithmetic ball quotients

Abstract: We study complex hyperbolic lattices that are not necessarily arithmetic. We prove that, if the associated ball quotient contains infinitely many maximal totally geodesic subvarieties, then the lattice is arithmetic. The idea is to realise such quasi-projective varieties inside a period domain for polarised integral Hodge structures and interpret totally geodesic subvarieties as unlikely intersections. Our theorem is indeed a special case of Klingler’s generalised Zilber-Pink conjecture. If time permits we will also discuss André-Oort type conjectures for non-arithmetic ball quotients. This is joint work with Emmanuel Ullmo.

Title: Arithmetic statistics via graded Lie algebras

Abstract: In work with Jack Thorne, we find the average size of the 3-Selmer group for a family of genus-2 curves by analyzing a graded Lie algebra of type E_8. In this talk, I will give examples of graded Lie algebras and show that they naturally arise when looking at families of algebraic curves. I'll talk about the role Lie theory plays in my work with Thorne, and about a new construction that extends our methods.

**Tuesday the 12th of May. **

**Schedule:**

14:00 - 15:00 Omri Faraggi (KCL) Title: Models of curves

15:30 - 16:30 Samir Siksek (Warwick) Title: Unit equations and the asymptotic Fermat conjecture.

**Abstracts:**

Title: Models of curves

Abstract: Let C be a smooth curve over a discretely valued field K. We

would like to study the arithmetic of C using models of C-flat O_K

schemes whose generic fibre is isomorphic to C. Of particular interest

are regular models – models where the ambient scheme is regular. These

are theoretically always possible to compute using blow ups, but this is

an expensive and time-consuming process. In this talk, we shall discuss

effective methods to calculate regular models, focussing in particular

on calculating regular models of hyperelliptic curves C : y^2 = f(x)

using so-called cluster pictures. These are a rather recent innovation,

capturing the combinatorial information of the roots of f and the p-adic

distances between them. If time allows, we shall discuss strategies for

general genus 3 curves as well.

Title: Unit Equations and the Asymptotic Fermat Conjecture

Abstract: The asymptotic Fermat conjecture (AFC) asserts that given a

number field K not containing the primitive third roots of unity, then

there is a constant B_K so that for all prime exponents l>B_K the

Fermat equation x^l+y^l+z^l=0 has no non-trivial solutions

in K. Recent work by Freitas, Kraus and Siksek has uncovered an

unexpected connection between AFC and S-unit equations. We give an

overview of this and the resulting progress on AFC over

Z_p-extensions.

**Tuesday the 28th of January. **

**Schedule:**

13:00 Daniel Loughran (Bath) Title: Hasse principle for a family of K3 surfaces

14:30 Cecília Salgado (MPIM Bonn/UFRJ) Title: Mordell Weil rank jumps and the Hilbert property

16:00 Damian Rössler (Oxford) Title: Purely inseparable points on curves

**Abstracts:**

Title: Hasse principle for a family of K3 surfaces

Abstract: In this talk we study the Hasse principle for the family of "diagonal K3 surfaces of degree 2", given by the explicit equations:

w^2 = A_1 x_1^6 + A_2 x_2^6 + A_3 x_3^6.

I will explain how many such surfaces, when ordered by their coefficients, have a Brauer-Manin obstruction to the Hasse principle. This is joint work with Damián Gvirtz and Masahiro Nakahara.

Title: Mordell Weil rank jumps and the Hilbert property.

Abstract: Let X be an elliptic surface with a section defined over a number field. Specialization theorems by Néron and Silverman imply that the rank of the Mordell-Weil group of special fibers is at least equal to the MW rank of the generic fiber. We say that the rank jumps when the former is strictly large than the latter. In this talk, I will discuss rank jumps for elliptic surfaces fibred over the projective line. If the surface admits a conic bundle we show that the subset of the line for which the rank jumps is not thin in the sense of Serre. This is joint work with Dan Loughran.

Title: Purely inseparable points on curves

Abstract: We give effective upper bounds for the number of purely inseparable points on non isotrivial curves over function fields of positive characteristic and of transcendence degree one. These bounds depend on the genus of the curve, the genus of the function field and the number of points of bad reduction of the curve.

**Caring costs:**

There is some money available to subsidise caring costs of parents or carers wishing to attend. Send me, Chris or Martin an email in advance, as funds are limited.