Seminars in Diophantine Equations

Robert Wilms: On equidistribution in Arakelov theory

26th January, 2023

As a motivating example of its own interest I will first discuss a new equidistribution result for the zero sets of integer polynomials. More precisely, I will give a condition such that the zero sets tends to equidistribute with respect to the Fubini-Study measure and I will show that this condition is generically satisfied in sets of polynomials of bounded Bombieri norm. In the second part, I will embed this example in a much more general framework about the distribution of the divisors of small sections of arithmetically ample hermitian line bundles in Arakelov theory.

Alexander Gamburd: Varieties of Markoff Type: Arithmetic, Combinatorics, Dynamics

12th December, 2022

The Markoff equation x2+y2+z2=3xyz, which arose in his spectacular thesis (1879), is ubiquitous in a tremendous variety of contexts. After reviewing some of these, we will discuss (briefly) asymptotics of integer points, and (in some detail) recent progress towards establishing forms of strong approximation on varieties of Markoff type, as well as ensuing implications, Diophantine and dynamical.

Timothy Browning: Polynomials over ℤ and ℚ: counting and freeness

31st October, 2022

Humans have been thinking about polynomial equations over the integers, or over the rational numbers, for many years. Despite this, their secrets are tightly locked up and it is hard to know what to expect, even in simple looking cases. In this talk I’ll discuss recent efforts to understand the frequency of integer solutions to cubic polynomials, before turning to the much more evolved picture over the rational numbers.

William Goldman: Cubic surfaces and non-Euclidean geometry

24th January, 2022

The classification of geometric structures on manifolds naturally leads to actions of automorphism groups, (such as mapping class groups of surfaces) on 'character varieties' (spaces of equivalence classes of representations of surface groups).

Just as surfaces of Euler characteristic −1 and −2 are the building blocks of surfaces, their character varieties are the building blocks of character varieties in general. They enjoy natural mapping class group-invariant Posson structures.

The simplest examples are affine cubic surfaces, such as the Markoff surface

x2+y2+z2 = xyz

which parametrizes complete hyperbolic structures on the punctured torus. In general these actions are dynamically complicated.  Another notable example is the surface

x2+y2+z2−xyz = k+2,

where k = 18. This affine cubic relates to Clebsch's diagonal cubic surface

a3+b3+c3+d3+e3 = a+b+c+d+e = 0.

Here the dynamics bifurcates from ergodic (with respect to the Poisson measure) when 2 < k < 18 to wandering for k > 18. This bifurcation can be understood in terms of lines on Clebsch's surface.

Pierre Debes: The Hilbert-Schinzel specialization property

10th January, 2021

Hilbert's Irreducibility Theorem shows that irreducibility over the field of rationals is 'often' preserved when one specializes a variable in some irreducible polynomial in several variables. I will present a version 'over the ring' for which the specialized polynomial remains irreducible over the ring of integers. The result also relates to the Schinzel Hypothesis about primes in value sets of polynomials: I will discuss a weaker 'relative' version for the integers and the full version for polynomials. The results extend to other base rings than the ring of integers; the general context is that of rings with a product formula.

Jennifer Balakrishnan: A tale of three curves

16th July, 2020

We will describe variants of the Chabauty-Coleman method and quadratic Chabauty to determine rational points on curves. In so doing, we will highlight some recent examples where the techniques have been used: this includes a problem of Diophantus originally solved by Wetherell and the problem of the "cursed curve", the split Cartan modular curve of level 13. This is joint work with Netan Dogra, Steffen Mueller, Jan Tuitman, and Jan Vonk.

Peter Sarnak: Integer points on affine cubic surfaces

25th June, 2020

The level sets of a cubic polynomial in four or more variables tends to have many integer solutions, while ones in two variables a limited number of solutions. Very little is known in case of three variables. For cubics which are character varieties (thus carrying a nonlinear group of morphisms) a Diophantine analysis has been developed and we will describe it. Passing from solutions in integers to integers in say a real quadratic field there is a fundamental change which is closely connected to challenging questions about one-commutators in SL2 over such rings.

Timothy Browning: Random Diophantine equations

4th June, 2020

I’ll survey some of the key challenges around the solubility of polynomial Diophantine equations over the integers. While studying individual equations is often extraordinarily difficult, the situation is more accessible if we merely ask what happens on average and if we restrict to the so-called Fano range, where the number of variables exceeds the degree of the polynomial. Indeed, about 20 years ago, it was conjectured by Poonen and Voloch that random Fano hypersurfaces satisfy the Hasse principle, which is the simplest necessary condition for solubility. After discussing related results I’ll report on joint work with Pierre Le Boudec and Will Sawin where we establish this conjecture for all Fano hypersurfaces, except cubic surfaces.

Andrew Sutherland: Sums of three cubes

7th May, 2020

In 1953 Mordell asked whether one can represent 3 as a sum of three cubes in any way other than 13+13+13 and 43+43 -53. Mordell's question spurred many computational investigations over the years, and while none found a new solution for 3, they eventually determined which of the first 100 positive integers k can be represented as a sum of three cubes in all but one case: k=42. In this talk I will present joint work with Andrew Booker that used Charity Engine's crowd-sourced compute grid to affirmatively answer Mordell's question, as well as settling the case k=42. I will also discuss a conjecture of Heath-Brown that predicts the existence of infinitely many more solutions and also explains why they are so difficult to find.