86 research outputs found
Model theory of special subvarieties and Schanuel-type conjectures
We use the language and tools available in model theory to redefine and
clarify the rather involved notion of a {\em special subvariety} known from the
theory of Shimura varieties (mixed and pure)
Special subvarieties of non-arithmetic ball quotients and Hodge Theory
Let be a lattice, and the
associated ball quotient. We prove that, if contains infinitely many
maximal totally geodesic subvarieties, then is arithmetic. We also
prove an Ax-Schanuel Conjecture for , similar to the one recently
proven by Mok, Pila and Tsimerman. One of the main ingredients in the proofs is
to realise inside a period domain for polarised integral variations
of Hodge structures and interpret totally geodesic subvarieties as unlikely
intersections
Applications of the hyperbolic Ax-Schanuel conjecture
In 2014, Pila and Tsimerman gave a proof of the Ax-Schanuel conjecture for the j- function and, with Mok, have recently announced a proof of its generalization to any (pure) Shimura variety. We refer to this generalization as the hyperbolic Ax-Schanuel conjecture. In this article, we show that the hyperbolic Ax-Schanuel conjecture can be used to reduce the Zilber-Pink conjecture for Shimura varieties to a problem of point counting. We further show that this point counting problem can be tackled in a number of cases using the Pila- Wilkie counting theorem and several arithmetic conjectures. Our methods are inspired by previous applications of the Pila-Zannier method and, in particular, the recent proof by Habegger and Pila of the Zilber-Pink conjecture for curves in abelian varieties
The theory of the exponential differential equations of semiabelian varieties
The complete first order theories of the exponential differential equations
of semiabelian varieties are given. It is shown that these theories also arises
from an amalgamation-with-predimension construction in the style of Hrushovski.
The theory includes necessary and sufficient conditions for a system of
equations to have a solution. The necessary condition generalizes Ax's
differential fields version of Schanuel's conjecture to semiabelian varieties.
There is a purely algebraic corollary, the "Weak CIT" for semiabelian
varieties, which concerns the intersections of algebraic subgroups with
algebraic varieties.Comment: 53 pages; v3: Substantial changes, including a completely new
introductio
Ax-Schanuel for Shimura varieties
We prove the Ax-Schanuel theorem for a general (pure) Shimura variety
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