Arithmetic of the j-invariant – JINVARIANT

The j-invariant is one of the most important and intriguing mathematical objects. In this project we will focus on studying arithmetical properties of the j-invariant, its generalizations, and adjacent objects (modular curves, modular forms, singular moduli, etc.). The members of the project are 6 French researchers. Two PhD students will be involved as well.

Our research plan splits into 4 big tasks:.

A. Arithmetic of Modular Curves.
The effective development of M. Kim's ideas about non-abelian Chabauty has been a major breakthrough in the arithmetic of curves, with particularly strong effects on modular curves, like the recent spectacular work of Balakrishnan et al. The situation is still evolving quite fast, as a new, more geometric formulation of the method, has been developped by Edixhoven and Lido. Project A will endeavour to develop both theoretical tools and practical investigations about that point of view (description of regular models for curves, generalization of the method to quotient abelian varieties, computations of new and (hopefully) meaningful examples).

B. Effective André-Oort.
In 2011 Pila proved the André-Oort Conjecture for C^n. His proof was non-effective, but in the recent years, effective proofs were given for various special cases of Pila's result. In particular, it is proved that André-Oort is effective for curves and for linear varieties (in other words, for algebraic varieties of dimension 1 or of degree 1). Another kind of results obtained is complete characterization of special points on certain infinite families of curves.
We plan to continue this activity. The ultimate goal is full effective André-Oort for C^n. While this can be hard to achieve, we plan to do many special cases, going far beyond degree and dimension one. We also plan to extend the fully explicit results on parametric families to higher dimension.

C. Singular Units and Drinfeld Modules.
The arithmetic of singular moduli, i.e. j-invariants of elliptic curves with complex multiplication, has recently achieved striking progress about units among these moduli, with the work of Habegger (2015), and Bilu, Habegger, Kühne (2018). We will investigate similar but unexplored questions for singular moduli attached to Drinfeld modules over function fields of positive characteristic.

D. Arakelov Geometry of Modular and Shimura Curves: we will study the self-intersection of their relative dualizing sheaf.
In Ullmo’s proof of Bogomolov’s conjecture, a principal ingredient was showing that the self-intersection product of the relative dualizing sheaf of an algebraic curve is strictly positive. Due to the work of Szpiro, a possible approach to make this result effective is obtaining estimates for the self-intersection product, as did Abbes and Ullmo for modular curves associated to Borel subgroups. In task D, we are interested in such estimates for the modular curves associated to normaliser of split and non split Cartan subgroups and some Shimura curves.

We will organize 2 conferences and 4 workshops on the topics of this project.