The Mordell conjecture 100 years later

I present the history of the proof of Mordell in 1983 and report on recent developments.

It is a fundamental question in mathematics to find rational solutions to a given system of polynomials, and in modern language this question translates into finding rational points in algebraic varieties. This question is already very deep for algebraic curves defined over ℚ. An intrinsic natural number associated with the curve, called its genus, plays an important role in studying the rational points on the curve. In 1983, Faltings proved the famous Mordell Conjecture (proposed in 1922), which asserts that any curve of genus at least 2 has only finitely many rational points. Thus the problem for curves of genus at least 2 can be divided into several grades: finiteness, bound, uniform bound, effectiveness. An answer to each grade requires a better understanding of the distribution of the rational points. In my talk, I will explain the historical and recent developments of this problem according to the different grades. I will also mention an ongoing work (joint with Shouwu Zhang) about a non-vanishing property and a Northcott property of the Gross-Schoen cycles and the Ceresa cycles for curves.

Refining the arguments of Dimitrov–Gao–Habegger and Kühne, we have been able to prove the general case of the uniform Mordell–Lang conjecture in joint work with Gao and Ge. This conjecture states that the intersection of a variety with a finite rank subgroup in an abelian variety is a union of finitely many cosets, whose number can be bounded from above purely in terms of the degree of the subvariety, irregardless of the ambient abelian variety. Our result also contains the full uniform Bogomolov conjecture in abelian varieties. In my talk, I will focus on our strategy to deal with higher-dimensional subvarieties.

In the past few years, the uniform Mordell conjecture proposed by Mazur was proved by Dimitrov–Gao–Habegger, and Kuhne, where the proof is based on Vojta's proof of the Mordell conjecture. By the theory of adelic line bundle of Yuan-Zhang, I have obtained a different proof which works over both number fields and function fields. In this talk, I will give a survey on these results.

In 1929 an epoch-making paper of Siegel offered a theorem representing the Mordell conjecture for affine curves: "A hyperbolic affine curve has only finitely many integral points" (which is a best-possible condition). In the talk I shall briefly survey on the history of this result, pointing out some proof-approaches, extensions to higher dimensions, and related issues, like effectivity or uniform estimates for the number of solutions of diophantine equations in integers.

The Mordell Conjecture (proved by Faltings in 1983) is a landmark result exemplifying the philosophy "Geometry controls arithmetic". However, it also implies that we can never hope to understand the arithmetic of a higher genus curve solely by studying its rational points over a fixed number field. In this talk, we will introduce the concepts of parametrized points and density degree sets and show how they, together with the Mordell-Lang conjecture, allow us to organize all algebraic points on a curve.

I will discuss some past, present, and hopes for future contacts between model theory and diophantine geometry. Such contacts are always mediated by a `tame' intermediate theory, such as differential or separably closed fields, difference fields, or the theory of the real exponential field with local analytic functions. I will describe the state of our attempts to construct such a tame theory capable of defining heights.

I will give a brief historical overview of the Mordell conjecture over function fields of positive characteristic (mainly). Then, I'll discuss its relationship with descent obstructions for rational points and present a recent result obtained with B. Creutz, which completes my earlier work with B. Poonen, showing that finite descent obstructions fully characterize the rational points on non-isotrivial curves over function fields.

I will present an overview of some explicit methods in the context of the Manin-Mumford and Mordell Conjecture. This allows to give examples of families of curves whose rational and torsion points can be explicitly determined.

Given a bounded submanifold M in ℝⁿ, how many rational points with common bounded denominator are there in a small thickening of M? Under what conditions can we count them asymptotically as the size of the denominator goes to infinity? I will discuss some recent work in this direction and arithmetic applications such as Serre's dimension growth conjecture as well as applications in Diophantine approximation. For this I'll focus on joint work with Shuntaro Yamagishi, as well as joint work with Rajula Srivastava and Niclas Technau.

Arithmetic dynamics is a relatively new field that draws inspiration from both algebraic geometry and classical complex dynamics. On its Diophantine side, arithmetic dynamics comprises a number of analogues and generalizations of statements concerning rational points, such as the Uniform Boundedness Conjecture of Morton and Silverman. In the reverse direction, ideas drawn from dynamical systems have deep implications for problems concerning rational points on curves and abelian varieties, particularly via potential theory and recent developments in Arakelov theory. This talk will discuss these two connections, along with the salient techniques that have emerged as well as avenues for further exploration.

Campana points are an arithmetic notion, first studied by Campana and Abramovich, that interpolates between the notions of rational and integral points. Campana points are expected to satisfy suitable analogs of Mordell's conjecture, Lang's conjecture, Vojta's conjecture and Manin's conjecture, and their study introduces new number theoretic challenges of a computational nature. In this talk I introduce Campana points, recall the main results, and discuss a range of open questions.

Over the last twenty-odd years, Minhyong Kim's method of non-abelian Chabauty has provided a powerful tool for addressing Mordell-like Diophantine problems. In this talk, I will give an overview of the development of the method: highlighting some of its key successes, and drawing out some common themes in the techniques used and the problems addressed. Time permitting, I will also say a few words about where I think the method is heading, and what I think the major challenges are.

I will give a short introduction to interactive theorem proving (using Lean) and will then discuss what would be needed to (a) state and (b) give a proof of Mordell's Conjecture in Lean, building on its mathematical library Mathlib.