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.

We prove some qualitative results about the $p$-adic Jacquet--Langlands correspondence defined by Scholze, in the $GL(2,Q_p)$, residually reducible case, by using a vanishing theorem proved by Judith Ludwig. In particular, we show that in the cases under consideration the $p$-adic Jacquet--Langlands correspondence can also deal with principal series representations in a non-trivial way, unlike its classical counterpart. The paper is available at http://arxiv.org/abs/1804.07567

Une question fondamentale mais difficile dans la théorie analytique des formes automorphes est la suivante : étant donné un groupe réductif $G$ et une représentation $r$ de son groupe-$L$, combien y a-t-il de représentations automorphes de conducteur analytique borné ? Dans cet exposé, je présenterai une réponse à cette question dans le cas où $G$ est un tore sur un corps de nombres.

We investigate, in characteristic 0, the compatibilities of various definitions of relatively unipotent log-de Rham fundamental groups for certain proper log-smooth integral morphisms of log-schemes. All the definitions are algebraic: this will allow us to give a purely algebraic proof of the p-adic good reduction criterium for curves (as given by Andreatta-Iovita-Kim). This is a joint work with Di Proietto and Shiho.

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.

There are several conjectures about K3 surfaces and abelian varieties of bounded dimension defined over number fields of bounded degree. They state that there are only finitely many possibilities for (1) the endomorphism algebra of an abelian variety, (2) the Neron-Severi lattice of a K3 surface, and (3) the Galois invariant subgroup of the geometric Brauer group. I will explain how these conjectures are related and what is known about them. This is a joint work with Martin Orr and Yuri Zarhin.

Le 12ème problème de Hilbert nous met au défi de construire des extensions abéliennes des corps de nombres. Je discuterai une construction $p$-adique pour les corps quadratiques réels, en utilisant des invariants qui se comportent comme étants des « valeurs spéciales de $j(z)$ ». Ceci est un travail en commun avec Henri Darmon.

Selon une remarquable conjecture de R. Pink, qui généralise celles de Manin-Mumford, Mordell-Lang, André-Oort, et de Bombieri-Masser-Zannier et Zilber, une sous-variété irréductible $Y$ d'une variété de Shimura mixte $S$ ne peut rencontrer de façon Zariski-dense la réunion des sous-variétés spéciales de $S$ de codimension $> \dim(Y)$ que si elle est contenue dans une sous-variété spéciale stricte de $S$ . On explicitera cette conjecture, et on la vérifiera, quand $S$ est le torseur de Poincaré sur une courbe elliptique à multiplications complexes. Il s'agit d'un travail avec H. Schmidt (ArXiv 1803.04835), et avec B. Edixhoven.

Elliptic curves over the rationals admit maps from various Shimura curves, and the comparison ratio of the degrees of these maps recovers important information on $abc$-triples. On the other hand, this ratio can be controlled by the Arakelov height of CM points. This requires a number of tools: zero-density estimates for L-functions, integral models for various objects, Galois representations, and some complex-analytic estimates. The final outcome is an unconditional estimate for the product of $p$-adic valuations of $abc$-triples, which lies beyond the reach of existing methods in the context of the $abc$ conjecture such as linear forms in logarithms. Our methods also yield other results. For instance, for totally real fields $F$ of bounded degree, we prove that the Faltings height of modular elliptic curves $E$ over $F$ is bounded linearly on log(modular degree of $E$) + log(Disc. of $F$). The logarithmic dependence of the discriminant of $F$ can be seen as evidence towards Vojta’s conjecture on algebraic points of bounded degree.

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