Séminaire de théorie des nombres de l'IMJ-PRG

We focus on a datum of a projective variety $X$ over a global field $K$ and an ample and semipositive adelically metrized line bundle L, such that all of Zhang's successive minima are equal (to 0, without loss of generality). In other words: the associated height function on the $\overline{K}$-points of $X$ is assumed to be non-negative and also to realize the infimum 0 as a limit value under some Zariski-generic sequence. Since the basic example is given by Rumely's capacitary height functions on the projective line, where the Chern form of $L=O(1)$ is the harmonic measure on a compact planar set, we propose to call such a datum a "harmonic arithmetic variety." We shall start by outlining a proof that the set of harmonic subvarieties of $X$ is stable under taking an intersection. For reasons that shall be explained, this result can be considered as an abstract generalization of the Bogomolov conjecture, in whose background there is no algebraic group or dynamical system present. We shall conclude by stating some further results, problems and conjectures concerning harmonic arithmetic varieties and the distribution of their algebraic points with respect to the natural height function.

Hasse et Weil conjecturent que les fonctions Zeta des variétés définies sur les corps de nombres admettent un prolongement méromorphe au plan complexe et satisfont une équation fonctionnelle. Pour les courbes de genre 1 sur les rationnels, cela résulte des travaux de Wiles et Breuil, Conrad, Diamond, Taylor qui expriment la fonction Zeta à l'aide d'une forme modulaire de poids $2$ ... On expliquera comment prouver un résultat analogue pour les courbes de genre $2$ en utilisant des formes modulaires sur des groupes de rang supérieur. Travail en commun avec G. Boxer, F. Calegari, T. Gee.

We present three novel algorithms to factor polynomials in one variable over finite fields using the arithmetic of Drinfeld modules. The first algorithm estimates the degree of an irreducible factor of a polynomial from Euler-Poincare characteristics of random Drinfeld modules. Knowledge of a factor degree allows one to rapidly extract all factors. The second algorithm is a random Drinfeld module analogue of Berlekamp's algorithm, partly inspired by Lenstra's elliptic curve method for integer factorization. The third algorithm employs Drinfeld modules with complex multiplication and will be the primary focus of the talk. The main idea is to compute a lift of the Hasse invariant with Deligne's congruence playing a critical role. We will discuss practical implementations and complexity theoretic implications of the algorithms.

Dans cet exposé, nous introduirons la notion de somme d’Apéry multiple et montrerons que tout multizêta peut s’exprimer comme une combinaison Z-linéaire de telles sommes. Il y a même une manière canonique de le faire. Cela place dans un contexte théorique unifié de nombreuses identités éparses dans la littérature et fournit un moyen systématique d’en engendrer de nouvelles dont certaines, étonnamment simples, n’avaient pas été découvertes auparavant.

Nous expliquerons comment l'étude des sommes de Gál permet d'obtenir des informations nouvelles concernant des valeurs de la fonction zêta de Riemann sur la droite critique. Ce travail a été réalisé en collaboration avec Gérald Tenenbaum et précise notamment les travaux de Bondarenko et Seip.

Moduli spaces of global $\mathbb G$-shtukas play a crucial role in the Langlands program over function fields. We analyze their functoriality properties concerning a change of the curve and a change of the group scheme $\mathbb G$ under various aspects. In the end we discuss a potential application to a proof of the non-emptiness of minimal KR-strata.

Anabelian geometry with étale homotopy types generalizes in a natural way classical anabelian geometry with étale fundamental groups. We show that, both in the classical and the generalized sense, any point of a smooth variety over a field $k$ which is finitely generated over $\mathbb{Q}$ has a fundamental system of (affine) anabelian Zariski-neighbourhoods. This was predicted by Grothendieck in his letter to Faltings. (Joint work with J. Stix)

Euler systems are compatible families of Galois cohomology classes attached to a global Galois representation, and they play an important role in proving cases of the Bloch—Kato conjecture. In my talk, I will explain the construction both of an Euler system and of a $p$-adic $L$-function attached to the spin representation of a genus $2$ Siegel modular form. I will also sketch a strategy for proving an explicit reciprocity law, relating the bottom class of the Euler system to values of the $p$-adic $L$-function. This is work in progress with David Loeffler, Vincent Pilloni and Chris Skinner.

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We construct infinite-dimensional continuous-time dynamical systems attached to integral normal schemes which are flat and of finite type over the spectrum of the integers. We study the periodic orbits and connectedness properties of these systems and ask several questions.

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