123 research outputs found
Ax-Schanuel for Shimura varieties
We prove the Ax-Schanuel theorem for a general (pure) Shimura variety
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 condition in arbitrary characteristic
We prove a positive characteristic version of Ax's theorem on the
intersection of an algebraic subvariety and an analytic subgroup of an
algebraic group. Our result is stated in a more general context of a formal map
between an algebraic variety and an algebraic group. We derive transcendence
results of Ax-Schanuel type
O-minimality and certain atypical intersections
We show that the strategy of point counting in o-minimal structures can be
applied to various problems on unlikely intersections that go beyond the
conjectures of Manin-Mumford and Andr\'e-Oort. We verify the so-called
Zilber-Pink Conjecture in a product of modular curves on assuming a lower bound
for Galois orbits and a sufficiently strong modular Ax-Schanuel Conjecture. In
the context of abelian varieties we obtain the Zilber-Pink Conjecture for
curves unconditionally when everything is defined over a number field. For
higher dimensional subvarieties of abelian varieties we obtain some weaker
results and some conditional results
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