Model theory of special subvarieties and Schanuel-type conjectures

Abstract

We use the language and tools available in model theory to redefine and clarify the rather involved notion of a special subvariety known from the theory of Shimura varieties (mixed and pure).

Bi-algebraic geometry and the André-Oort conjecture

“…In this section we introduce a general format in which a Manin-Mumford-André-Oort type problem can be formulated: the notion of a special structure on an algebraic variety. We refer to [U16] for more details and [Zil13] for a study of special subvarieties from the point of view of model theory.…”

Canonical models of modular curves and the Galois action on CM-points

“…See e.g. [9] or [10] for a survey on this project. In particular, we are interested in [11] in the situation that arises in the context of a system of analytic functions…”

Ax–Schanuel for linear differential equations

“…Since (1.2) holds for pseudo-exponentiation (it is included in the axiomatisation given by Zilber), Zilber's conjecture implies Schanuel's conjecture. For details on pseudo-exponentiation see [Zil04,Zil05,Zil02,Zil16]. Though Schanuel's conjecture seems to be out of reach of modern methods in mathematics, James Ax proved its differential analogue in 1971 ( [Ax71]).…”