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).

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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.

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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.

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).

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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.

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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.

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