Andre Oort Research Articles

An explicit Andre--Oort type result for $\mathbb{P}^1(\mathbb{C}) \times \mathbb{G}_m(\mathbb{C})$ based on logarithmic forms

Using linear forms in logarithms we prove an explicit result of Andre Oort type for P-1(C) x G(m)(C). In this variation the special points of P-1(C) x G(m)(C) are of the form (alpha, lambda), with a a singular modulus and lambda a root of unity. The qualitative version of our result states that if C is a closed algebraic curve in P-1(C) x G(m)(C), defined over a number field, not containing a horizontal or vertical line, then C contains only finitely many special points. The proof is based on linear forms in logarithms. This differs completely from the method used by the author recently in the proof of the same kind of statement, where class field theory was applied.

An explicit André–Oort type result for

AbstractUsing class field theory we prove an explicit result of André–Oort type for $\mathbb{P}^1(\mathbb{C}) \times \mathbb{G}_m(\mathbb{C})$. In this variation the special points of $\mathbb{P}^1(\mathbb{C})$ are the singular moduli, while the special points of $\mathbb{G}_m(\mathbb{C})$ are defined to be the roots of unity.

Bounded equidistribution of special subvarieties in mixed Shimura varieties

In this paper, we study the equidistribution of certain families of special subvarieties in general mixed Shimura varieties. We introduce the notion of bounded sequences of special subvarieties, and we prove that the Andre–Oort conjecture holds for such sequences. The proof follows the equidistribution approach used by Clozel, Ullmo, and Yafaev in the pure case. We then propose the notion of test invariant of a special subvariety, which is adapted from the lower bound formula of degrees of special subvarieties in the pure case studied by Ullmo and Yafaev, and we show that sequences of special subvarieties with bounded test invariants are bounded, hence the Andre–Oort conjecture holds in this case.

Towards the Andre–Oort conjecture for mixed Shimura varieties: The Ax–Lindemann theorem and lower bounds for Galois orbits of special points

Abstract We prove in this paper the Ax–Lindemann–Weierstraß theorem for all mixed Shimura varieties and discuss the lower bounds for Galois orbits of special points of mixed Shimura varieties. In particular, we reprove a result of Silverberg [57] in a different approach. Then combining these results we prove the André–Oort conjecture unconditionally for any mixed Shimura variety whose pure part is a subvariety of 𝒜 6 n {\mathcal{A}_{6}^{n}} and under the Generalized Riemann Hypothesis for all mixed Shimura varieties of abelian type.

On special subvarieties of Kuga varieties

In this paper we prove the equidistribution of certain families of special subvarieties in Kuga varieties, which is a special case of the general Andre–Oort conjecture formulated for mixed Shimura varieties. Our approach is parallel to the pure case treated in the works of L. Clozel, E. Ullmo, and A. Yafaev, which uses tools from ergodic theory.

An unconditional proof of the André–Oort conjecture for Hilbert modular surfaces

We give an unconditional proof of the Andre–Oort conjecture for Hilbert modular surfaces asserting that an algebraic curve contained in such a surface and containing an infinite set of special points, is special. The proof relies on a combination of Galois-theoretic techniques and results from the theory of o-minimal structures.

Zariski-density of exceptional sets for hypergeometric functions

We extend the results of Desrousseaux in [10], [11], [12]. In those papers an exceptional set was constructed for the Appell– Lauricella hypergeometric functions associated to rational ball tuples. The definition of this set requires quite subtle conditions necessary for the application of the Andre–Oort conjecture. In this paper, we show that generalizations of the Andre–Oort conjecture by Pink [20] lead to similar results for a more natural exceptional set, namely the set of algebraic points at which the function takes algebraic values. Desrousseaux’s exceptional set is in general a proper subset of this set.