Schedule for: 22w5162 - Modern Breakthroughs in Diophantine Problems

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

We say a degree $d$ point $x$ on a curve $C$ is isolated if it does not belong to an infinite family of degree $d$ points parametrized by a geometric object — either $\mathbb{P}^1$ or a positive rank abelian subvariety of the curve's Jacobian. We say $x$ is sporadic if there are only finitely many points on $C$ of degree at most $d$. Every sporadic point is isolated, but the converse need not hold. Motivated by problems concerning the arithmetic of elliptic curves, we will focus on the case where $C$ is the modular curve $X_1(N)$. By prior joint work with Ejder, Liu, Odumodu, and Viray, Serre's Uniformity Conjecture implies there are only finitely many elliptic curves with $j$-invariant in $\mathbb{Q}$ which give rise to an isolated point of any degree on $X_1(N)$, even as $N$ ranges over all integers. On the other hand, an analogous finiteness result on non-CM $\mathbb{Q}$-curves would actually imply Serre's Uniformity Conjecture, as shown in recent joint work with Najman. In this talk, I will discuss unconditional results in this direction for $\mathbb{Q}$-curves giving rise to sporadic points of odd degree (joint with Filip Najman) and highlight several remaining open questions along these lines of investigation.

Building on Mazur's 1978 isogeny theorem, Kenku in 1982 determined all integers which arise as the degree of a cyclic isogeny between rational elliptic curves. In this talk we explain Mazur's strategy to bounding all such integers from having bounded the prime degree isogenies, and we apply this method to find the first example of the determination of cyclic isogeny degrees for elliptic curves over a number field other than $\mathbb{Q}$. This is work in progress with Filip Najman and Oana Padurariu.

In this talk I will discuss recent work on some generalized Fermat equations of the form $x^2 + y^{2n} = z^p$. I will show how one can construct various Frey curves, both over the rationals and over totally real fields, to prove that certain infinite families of generalized Fermat equations have no non-trivial solutions in coprime integers. This strategy relies on modularity and level-lowering results over totally real fields, as well as computations with Hilbert cuspidal eigenforms. I will highlight the role played by various modular curves in obtaining these results.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Given an elliptic curve over a number field with its Weierstrass model, we can study the integral points on the curve. Taking an infinite family of elliptic curves and imposing some ordering, we may ask how often a curve has an integral point and how many integral points there are on average. We expect that elliptic curves with any non-trivial integral points are generally very sparse. In certain quadratic and cubic twist families, we prove that almost all curves contain no non-trivial integral points.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will explain joint work with Bhargava and Shnidman in which we show: 1. At least a sixth of integers aren’t the sum of two rational cubes, and 2. At least a sixth of odd integers are the sum of two rational cubes! (It’s easy that 0% of integers are the sum of two integral cubes!)

We give a new proof and then generalize a result of Greenberg and Kurihara about the field of definition of torsion points for quotients of Fermat curves.

We discuss a method for reducing models of curves equipped with a map to the projective line which is unramified away from three points. Using the ramified points of the map, we can compute "small" functions supported at these points to produce "small" plane models of the original curve. It is then possible to rescale a plane model in an optimal way to reduce the size of the coefficients, for which we use an integer linear program. We implemented and ran the algorithm on a database of Belyi maps in the LMFDB with often very favourable results. This is joint work with Sam Schiavone and John Voight.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Please think of some open problem to present.

An important invariant of a curve defined over a number field is the minimal degree for which it has infinitely many closed points of that degree. In this talk I will discuss joint work with Borys Kadets, extending classification results of Harris--Silverman and Abramovich--Harris, in which we characterize when this invariant takes small values.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk I'll sketch a proof of a Diophantine approximation result in the direction of Vojta's conjecture for rational points with truncated counting functions. The result essentially says that on a variety $X$ with an algebraic point $P$ and a divisor $D$ with sufficiently many components, the truncated counting function $N'(D,x)$ for rational points $x$ is bounded from below in terms of the $v$-adic proximity of $x$ to $P$ for any fixed place $v$. The method of proof is based on linear forms in logarithms and a geometric construction.

In this talk, I will discuss how machine learning can aid in the computation of the periods of projective hypersurfaces. I will also report on the results of our large scale computation to find the periods of all smooth quartics in $\mathbb{P}^3$ that are the sum of $5$ monomial terms with unit coefficients. Joint work with Kathryn Heal and Emre Sertoz.

I will report on recent developments that surround Schmidt's Subspace Theorem and the question of Diophantine approximation for rational points in projective varieties. These results are at the intersection of Diophantine arithmetic geometry and higher dimensional birational geometry (including the area of $K$-stability). As I will explain, a key tool is the filtration method of Ru and Vojta. It generalized and refined earlier work of Autissier, Corvaja-Zannier, Levin, Ru and others. After outlining the key points behind the construction, I will state a form of the resulting Arithmetic General Theorem. This formulation includes a description of the Diophantine exceptional set. There are consequences for points of bounded degree, given a suitable form of the Second Main Theorem. Further, I will explain how the General Theorem can be used, together with K. Fujita's valuative criteria for $K$-stability, the theory of Newton-Okounkov bodies and the Duistermaat-Heckman measure, to deduce instances of Vojta's Main Conjecture for $K$-unstable Fano varieties. Among other things, these results provide impetus for a concept of arithmetic $K$-instability and implications thereof. I will sketch the main ideas in this regard. Finally, I will discuss additional more recent and ongoing work which pertains to the twisted height functions and the parametric subspace theorem within the context of linear series and the Diophantine approximation sets of Schmidt. This viewpoint, in particular, expands on that of Evertse, Ferretti, Schlickewei and Schmidt.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

After recalling the analogue of Campana's conjecture over function fields, we will present some unconditional and effective partial progress on this problem. If time permits, we will work out our result in an explicit example.

In this talk I will give a brief overview of the modular approach for solving Diophantine equations over totally real number fields involving modularity, level lowering and image of inertia comparison along with the study of certain S-unit equations. We will see a few concrete examples of asymptotic results using this method for equations of signatures $(p,p,2)$ and $(p,p,3)$. If time permits, I will show how one can stretch out this method to study equations with signatures $(r,r,p)$ by using the so-called multi-Frey approach.

The ideas of Wiles on the modularity of elliptic curves over $\mathbb{Q}$, and subsequent extensions and adaptations, have had a great influence on the study of Diophantine equations through the modular method. There is an analogous concept for elliptic curves over function fields over finite fields, known as Drinfeld modularity: an elliptic curve over ${\mathbb F}_q(t)$ with split multiplicative reduction at infinity is covered by a Drinfeld modular curve, which parametrizes Drinfeld modules of rank $2$ with a suitable level structure. More generally, let $E$ be an elliptic curve over ${\mathbb F}_q(t)$, and let $E_i$ be the elliptic curve over ${\mathbb F}_q(t_1,\dots,t_n)$ obtained by replacing $t$ by $t_i$. Then there is an $n$-dimensional moduli space of "shtukas" over ${\mathbb F}_q(t_1,\dots,t_n)$ that is conjectured to be in correspondence with $E_1 \times \dots E_n$. We describe how to construct these moduli spaces concretely as the sets of $2 \times 2$ matrices of polynomials satisfying certain specialization conditions and prove the conjecture in a few special cases by means of computations on K3 surfaces.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In recent joint work with Josha Box and Stevan Gajović, we determined cubic and quartic points on a collection of modular curves whose Jacobians have positive rank. To do this we further developed Siksek's symmetric Chabauty method. In this talk, I will outline how this method works, how to apply it to explicit examples and point out complications that can arise when trying to do so.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.

Given an abelian scheme defined over $\bar{\mathbb{Q} }$and an irreducible subvariety $X$ which dominates the base, the Relative Manin-Mumford Conjecture (proposed by Zannier) predicts how torsion points in closed fibers lie on $X$. The conjecture says that if such torsion points are Zariski dense in $X$, then the dimension of $X$ is at least the relative dimension of the abelian scheme, unless $X$ is contained in a proper subgroup scheme. In this talk, I will present a proof of this conjecture. As a consequence this gives a new proof of the Uniform Manin-Mumford Conjecture for curves (recently proved by Kühne) without using equidistribution. This is joint work with Philipp Habegger.

Let $F$ be a homogenous irreducible form of degree at least $3$ with integer coefficients. A Thue-Mahler equation is a Diophantine equation of the form $$ F(X,Y) = a\cdot p_1^{z_1}\cdots p_v^{z_v}$ where $a$ is an integer and $p_1, \dots, p_v$ are rational primes. To explicitly solve these equations when the degree and set of primes is small, there exists a practical method of Tzanakis-de Weger using linear forms in logarithms. We give new lattice sieving techniques and a refined algorithm capable of resolving Thue-Mahler equations with high-degree and large prime sets. In this talk, we describe our refined implementation of this method and discuss the key steps used in our algorithm.

In 1998 Friedlander and Iwaniec proved that $x^2 + y^4$ represents infinitely many primes and produced an asymptotic formula for the number of such representable primes. Their work transformed analytic number theory forever. In 2017 Heath-Brown and Li refined their work by showing that $a^2 + p^4$ is prime infinitely often, with $p$ a prime variable. In this talk we generalize their results by showing that for any irreducible binary quadratic form $f$ satisfying $f(x,1) \not \equiv x(x+1) \pmod{2}$ we have that both $f(a,b^2)$ and $f(a,p^2)$ are prime infinitely often.