In 2012 L. Taelman proved a remarkable analytic class number formula that relates the special values of \(\infty\)-adic Goss \(L\)-functions (associated to Drinfeld modules) to the \(\infty\)-adic regulator of "units". In this talk we will present recent advances (due to various authors : D. Adam, X. Caruso, L. Denis, Q. Gazda, A. Lucas ...) related to specials values of \(v\)-adic Goss \(L\)-functions where \( v\) is a finite place. A part of this talk is based on a work in progress with V. Bosser (Caen University).

Serre's open image theorem famously states that the p-adic representation of an elliptic curve without complex multiplication is surjective for all sufficiently large primes p. We consider the analogous statement for abelian threefolds and provide an effective and algorithmic upper bound on the largest non-surjective prime. This is joint work with Davide Lombardo.

Let \(Y\) be a curve over a \(p\)-adic number field \(K\). In this talk I discuss an algorithm for computing the Weil-Deligne representation of \(Y\) from its stable reduction to characteristic \(p>0\). We discuss examples of superelliptic and plane curves, where it is known how to compute the stable reduction. This is joint work with Bruno, Do, Wewers, and Zhang.

The height of an algebraic number is a real-valued function that measures the "arithmetic complexity" of the number. While numbers of height zero are well understood, many questions remain open regarding numbers of small height. For example, a key question is whether a given infinite algebraic extension of the rationals contains numbers of arbitrarily small (and non-zero) height. This talk will focus on situations where the answer is positive and, in particular, on the following question: in fields where small points can “obviously” be found, do these points have always "good reasons" to be small? For instance, the field generated over the rationals by all roots of 2 contains some obvious points of very small height (0, small fractional powers of 2 multiplied by roots of unity). Does it contain other small points? A very particular case of a conjecture of Rémond suggests that the answer is no. Rémond's conjecture more generally concerns the saturated closure of subgroups of finite ranks in tori and abelian varieties defined over number fields. It remains widely open and generalizes several important problems, such as Lehmer's conjecture. Recently, Pottmeyer established a necessary group-theoretical condition for the conjecture to hold and proved it in the case of tori. I will present joint work with G. A. Dill, where we extend this result by showing that the condition is also satisfied for split semi-abelian varieties (joint work with G. A. Dill).

Given two abelian varieties over a number field \(K\), we say that they are quadratic twists if they become isogenous after taking a quadratic extension of the base field. We moreover say that they are (strongly) locally quadratic twists if their reduction modulo almost all primes of \(K\) (or base-change to almost all completions of \(K\)) are quadratic twists. Clearly, two abelian varieties that are globally quadratic twists will also be (strongly) locally quadratic twists. The converse is not necessarily true. In this talk I will give an overview of results and counterexamples, based on joint work with E. Ambrosi and F. Fité.

Siegel's finiteness theorem for integral points on affine curves of genus >0 is, in general, not effective. In a joint work with D. Lombardo and U. Zannier, we consider families of curves of genus 2 with a single point at infinity and show that in many such families there is a dense set of fibres for which the integral points can be effectively determined. Our method is based on a criterion of Bilu and the study of the torsion values of sections of certain doubly elliptic abelian schemes.

In the last quarter-century, intersections that are deemed to be “unlikely” for dimension reasons have been proved to deserve their name in various contexts, ranging from intersections with algebraic subgroups of powers of the multiplicative group to intersections with special subvarieties of moduli spaces of abelian varieties. In my talk, I will report on joint work with Francesco Gallinaro, where we show that, in a power of the multiplicative group, also intersections with algebraic subgroups that are deemed to be “likely” for dimension reasons deserve their name in the sense that they are almost never empty as soon as we assume a mild technical condition, satisfied for example by all algebraic curves which are not contained in a coset of a proper subtorus. This is also related to Zilber's Exponential Algebraic Closedness Conjecture.

Let \(A\) be an abelian variety defined over a field K. We study finite generation properties of the profinite group \(Gal(K_\text{tor}(A)/K)\) and of certain closed normal subgroups thereof, where \(K_\text{tor}(A)\) is the torsion field of \(A\) over \(K\). In fact, we establish more general finite generation properties for monodromy groups attached to smooth projective varieties via étale cohomology. We apply this in order to give an independent proof and generalizations of a result of Checcoli and Dill about small exponent subfields of \(K_\text{tor}(A)/K\) in the number field case. We also give an application of our finite generation results in the realm of permanence principles for varieties with the weak Hilbert property. This is a report on a joint work with Sebastian Petersen.

The original problem is to find natural integers which are product of two consecutive integers and product of three integers consecutive in an arithmetic progression with common difference "a". We study in fact the set of integral points on the elliptic curve \(y^2+y=x^3-a^2x\) where \(a\) is a fixed positive integer. The problem can be studied via factorisation in the relevant cubic fields for certain values of a (Mordell, Godinho-Porto-Togbé, Lee-Louboutin). We show that using Diophantine arithmetic of elliptic curves (Archimedian uniformisation, reduction modulo p, Néron-Tate heights, linear forms in elliptic logarithms) completed with computer assisted calculations, yields a much clearer and complete picture: the rank of the elliptic curve is at least 2 (for a greater or equal to 2) and when the rank is two we obtain a complete description of the set of integral points. Joint work with Hemar Godinho and Diego Marques (UnB, Brasil).

For an elliptic curve \(E/ \mathbb{Q}\), computing its Tate-Shafarevich group is a fundamental but hard problem. Assuming its finiteness, the Birch and Swinnerton-Dyer conjecture (BSD) provides a formula for its size in terms of the special value of the L-function of \(E/\mathbb{Q}\) and some arithmetic invariants of the curve. If \(K/\mathbb{Q}\) is a Galois extension, one can ask if a BSD-like formula can be used to link the Galois module structure of the Tate-Shafarevich group of \(E/K\) to the special value of the twisted L-function of \(E/\mathbb{Q}\) by an Artin representation associated to \(K/\mathbb{Q}\). In this talk, I will present a joint work with Himanshu Shukla, where we give an explicit conjecture for such a link in the case of primitive Dirichlet characters and provide numerical verifications obtained via 11-descent procedures over \(C_5\)-extensions.

Isolated points on curves are, informally, points whose existence is not explained by the geometry of the curve. There has been considerable recent progress in understanding isolated points on modular curves. In this talk, we will survey some known results and sketch the proofs of several new ones. We also introduce the notion of twist isolated points and describe the advantages this framework has for describing unexpected rational points on modular curves. This is joint work with Maarten Derickx.

The Mahler measure of a polynomial is one of the most basic forms of height. In this talk, we will explore its various links with Iwasawa theory of curves, knots and graphs (based on joint work with Daniel Vallières), with dynamical systems and the analytic properties of the toric locus of a polynomial (based on joint work with François Brunault, Antonin Guilloux and Mahya Mehrabdollahei) and, most importantly, with special values of L-functions (based on joint work with François Brunault).

I'll discuss two recent results whose proofs use Selmer groups of hyperelliptic curves. The first is our recent resolution (with Alpoge, Bhargava, and Ho) of Hilbert's tenth problem over any ring of integers \(\mathcal{O}_K\), showing that there is no algorithm to determine solubility of diophantine equations over \(\mathcal{O}_K\). The second result (with Siad) is a rare instance of the Cohen-Lenstra-Martinet heuristics, in this case for cubic Kummer extensions. Our proof leads to a refined heuristic for the behavior of the 2-part of the class group in families of odd degree number fields.

Define, for a positive integer \(d\), \(S(d)\) to be the set of all primes \(p\) that occur as the order of a point \(P \in E(K)\) on an elliptic curve \(E\) defined over a number field \(K\) of degree \(d\). We discuss how some plausible conjectures on the sparsity of newforms with certain properties would allow us to deduce a fairly precise result on the asymptotic behavior of \(\max S(d)\) as \(d\) tends to infinity. This is joint work with Maarten Derickx.

We study a complex abelian scheme equipped with a section, analyzing the group of algebraic sections via the notion of abelian logarithm. While such logarithms are locally defined on the base, the obstruction to their global extension is captured by the integral relative monodromy group associated with the section. We establish results concerning the size of this monodromy group, with particular attention to its connection to the ramification behavior of the section. Under suitable hypotheses, the non-triviality of the relative monodromy is shown to be equivalent to the section being non-torsion. This is joint work with P. Dolce (Westlake University) and extends earlier results of Corvaja and Zannier in the context of elliptic surfaces. Related questions have also been recently investigated by Y. André using Hodge-theoretic methods.