| Summary: | <p>We consider a number of Diophantine problems for (mixed) Shimura varieties. Specifically, we look at the multiplicative relations satisfied by special points of modular and Shimura curves. These problems are closely related to the André--Oort and Zilber--Pink conjectures, and we will resolve some special cases of these conjectures.</p>
<p>Let $Y$ be a modular or Shimura curve. Let $V \subset Y \times \mathbb{G}_\mathrm{m}$ be an algebraic correspondence defined over $\overline{\mathbb{Q}}$. For each $n \geq 1$, we prove that all multiplicative dependencies among $n$ $V$-images of special points of $Y$ belong to one of finitely many components of $V^n$ of a particular special kind.</p>
<p>We then strengthen this result in the case that $Y$ is the modular curve $Y(1) = \mathrm{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$ and $V$ is the graph of a (suitably generic) rational function. In this case, we prove that, for each $n \geq 1$, there are only finitely many $n$-tuples of distinct $V$-images of special points which are multiplicatively dependent and minimal for this property. The key ingredient in this strengthening is a functional multiplicative independence statement for modular functions, which we prove.</p>
<p>Finally, we give a complete classification of the triples of singular moduli which have rational product. We show that all such triples are ``trivial'' in a suitable sense. This establishes a completely explicit André--Oort statement for the family of cubic surfaces defined (in $\mathbb{C}^3$) by an equation $x_1 x_2 x_3 = \alpha \in \mathbb{Q}$.</p>
|
|---|