Heights, Modularity, Transcendence – HAMOT

In this project a multi-university team of researchers plans to attack a variety of challenging problems in Diophantine Geometry and Transcendence theory. They can be loosely partitioned into four tasks, Bogomolov-Lehmer problems, Diophantine geometry and transcendence, Zilber-Pink conjecture, Modularity and transcendence.

1.The first task deals with lower bounds for heights of points, and, more generally, of subvarieties on commutative group varieties. Two central problems here are that of Lehmer, dealing with lower bounds for heights of non-torsion points and subvarieties, and that of Bogomolov, which deals only with the points on some proper subvariety. Impressive results are already obtained here: the original Bogomolov conjecture is proved by Zhang and others, and Lehmer’s conjecture is proved “up to epsilon” for tori and abelian varieties with CM. Also, partial versions of Lehmer’s conjecture are obtained over the maximal abelian extension of rationals (Amoroso, Dvornicich, Zannier). Still, more is unknown in this fascinating domain than known. What can one say the spectrum (the set of all values) of the height function on a subvariety? How do the lower bounds in Bogomolov’s conjecture depend on various parameters? Can one attack Lehmer’s problem for general (non-CM) abelian varieties, and for semi-abelian varieties (here even the “correct” statement of Lehmer’s conjecture is not known)? How can one bound the height away from 0 in families of abelian varieties (Silverman-Lang conjecture, still open)? And what happens in characteristic p?

2.On the opposite side, one can ask about upper bounds for heights of points, and on the related and often more general “non-density” problems. In spite of a number of celebrated results like finiteness and non-density theorems of Siegel, Faltings, Vojta and others, this remains one of the most difficult parts of mathematics, with a number of extremely hard conjectures. A new breath was given to this domain with the brilliant idea of Corvaja and Zannier to apply to the non-density problems of Diophantine geometry the Subspace theorem of Schmidt ans Schlickewei. We plan to make progress towards the conjecture of Levin: integral points on a d-dimensional variety with d+2 ample divisors at infinity are not Zariski-dense. This direction is also related to some problems of transcendence, most notable the work of Adamczewski and Bugeaud on transcendence of automatic numbers. We shall also investigate the effective and numerical aspect, going back to the work of Baker and others. We are especially interested in applying effective methods of Diophantine geometry to modular curves, in the spirit of the recent work of Bilu and Parent.

3.Another bunch of the “bounded height / non-density” problems goes back to the ground-breaking work of Bombieri, Masser and Zannier on the intersection of curves on a multiplicative group with algebraic subgroups, and culminated with the general conjecture of Zilber-Pink: given a subvariety X of a commutative group variety A, the intersections X with proper algebraic subgroups of A is not Zariski-dense, with the “obvious” exceptions. We are going to study many cases of this difficult conjecture. This task is closely connected with the previous two.

4.Finally, we will focus on the recently discovered vast and deep connections between the theory of modular forms and transcendence. We would like to develop at once the general theory of algebraic independence and the theory of modular forms and hypergeometric functions with linear independence and algebraic independence purposes in mind. Part of the problems proposed in this task concern positive characteristic counterparts of classical conjectures of algebraic independence. The ideas and methods of this task are quite related to those of Task 2.