Séminaire de théorie des nombres de l'IMJ-PRG

Since the foundational work of Néron and Tate in the 60s, various conjectural definitions of the height pairing for homologically trivial algebraic cycles on smooth, projective varieties over a number field, or more general global fields, have been proposed by Belinson, Bloch, Gillet-Soulé and Schneider, among others. The related concept of height functions on sub-varieties was extensively developed by Faltings and, in the early 2000s, extended by Moriwaki to varieties defined over finitely generated extensions of $\mathbb{Q}$, also called arithmetic function fields. More recently, Bruno Kahn has used algebraic intersection theory to give a geometric construction of a ``refined" height pairing over the function field of a smooth, projective variety $B$, taking its values in $\textrm{Pic}(B)$. In this talk, I will present an Arakelov-geometric analogue of this construction over an arithmetic function field $K$, taking its values in the arithmetic Picard group of a normal arithmetic variety whose function field is $K$. The construction is in two parts: first, Kahn's definitions are extended and adapted to arithmetic varieties using the intersection theory on regular schemes of Gillet-Soulé, and then the Archimedean term is added by considering generalized Green currents satisfying some ``admissibility" conditions, inspired by the original work of Arakelov. If time allows, I will discuss the connection with Moriwaki's work and with the work of Brosnan and Pearlstein on the degeneration of the Archimedean height pairing.

In 1952, Mordell erroneously claimed that the surface $xyz = G(x,y)$ has infinitely many integral points for any $G \in \mathbb{Z}[x,y]$. I will discuss a classification for finiteness and Zariski density of (positive) integral points on these log K3 surfaces when the coefficients of $G$ are non-negative. This combines a dynamical argument proposed by Mordell, the theory of generalized cluster algebras, and the Killing–Cartan classification of semisimple Lie algebras. This talk is partially based on joint work with Antoine de Saint Germain.

I will talk about the recent proof of algebraicity of Stark units for real quadratic fields using a new set of identities for special values of Faddeev's modular quantum dilogatithm. The talk is based on recent joint work with Campbell Wheeler.

If $A$ is a $g$-dimensional principally polarized abelian variety over $\mathbb{C}$, it is not hard to show that the Hecke orbit of $A$ (the set of principally polarized abelian varieties which are isogenous to $A$) is dense in $\mathcal{A}_g(\mathbb{C})$. This fails in positive characteristic because Hecke orbits are confined to Newton strata, the loci of $\mathcal{A}_g$ where the Newton polygon of the abelian variety is constant. The Hecke orbit conjecture, now a theorem of Chai-Oort, thus asks whether Hecke orbits are dense in their respective Newton strata over an algebraically closed field of characteristic $p$. In the first part of the talk, we will explain Chai's strategy for proving the Hecke orbit conjecture, and we will present the recent results in $p$-adic monodromy that led to a proof, by D'Addezio and van Hoften, of the Hecke orbit conjecture for all Shimura varieties of Hodge type. In the second part we present a refinement of these $p$-adic monodromy techniques, with an application to the rigidity of Tate-linear varieties, which are central objects in Chai's work.

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