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Effective André-Oort

Yuri Bilu

On every Shimura variety certain algebraic subvarieties are called "special"; in particular, "special points" are special subvarieties of dimension 0. Intuitively, if the Shimura variety is viewed as a moduli space of some objects (like Abelian varieties or so), then the special subvarities are moduli spaces of the same objects with some additional structure.

The celebrated André-Oort Conjecture asserts, roughly speaking, that the Zariski closure of a set of "special points" is a "special subvariety"; equivalently, every algebraic subvariety of a Shimura variety may have at most finitely many maximal special subvarieties. This conjecture is proved by Klingler, Ullmo and Yafaev subject to GRH, and in many special cases unconditionally.

In particular, Pila (2011) proved unconditional André-Oort for Shimura varieties of modular type, that is, products of modular curves. Here every point is \((x_1,...,x_n)\), where each \(x_i\) is an elliptic curve with some additional structure. Special subvarieties are (roughly) defined by conditions of the type "there is a cyclic isogeny of given degree between \(x_i\) and \(x_j\)" or "\(x_i\) is a given curve with Complex Multiplication".

While Pila's argument (based on the ground-breaking idea of a previous work of Pila and Zannier) is very clever and beautiful, it is non-effective, though the use of Siegel-Brauer lower estimate for the class number.

In the recent years, in the work of Masser, Zannier, Kühne, Binyamini and others effective proofs were given for various special cases of Pila's result. Most of them are based on the beautiful "Tatuzawa trick", discovered by Kühne, which allows one to achieve effectiveness, in many cases, by replacing the Siegel-Brauer by the Siegel-Tatuzawa Theorem.

The purpose of my course is to give an introduction into this topic. I will restrict to the "Shimura variety" \(C^n\) (viewed as the product of \(n\) \(j\)-lines). The special subvarieties are (roughly) defined by equations of the type \(F_N(x_i,x_j)=0\) or \(x_i=\)(singular modulus), where \(F_N\) is the modular polynomial of level \(N\), and singular moduli are j-invariants of elliptic curves with CM. In particular, special points are those whose all coordinates are singular moduli.

No preliminary knowledge beyond university course of Algebra, Analysis and Number Theory is required. I will even define what Complex Multiplication is. I will also try to highlight ideas as clearly as possible, in many cases sacrificing generality to lucidity. Here is an approximate plan.

  1. Introductory material: Complex Multiplication, Class Field Theory etc.
  2. Definable sets and the (non-effective) Pila-Zannier argument.
  3. One-dimensional effective/explicit results on individual curves and families on curves.
  4. Multidimensional results: the "naive" approach and the Binyamini approach (using his effective estimates for definable sets).