6,087 research outputs found
Towards the Andr\'e-Oort conjecture for mixed Shimura varieties: the Ax-Lindemann theorem and lower bounds for Galois orbits of special points
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 in a different approach. Then combining these results we prove the
Andr\'e-Oort conjecture for any mixed Shimura variety whose pure part is a
subvariety of A_6^n.Comment: The arXiv version differs from the published versio
Counting and computing regions of -decomposition: algebro-geometric approach
New methods for -decomposition analysis are presented. They are based on
topology of real algebraic varieties and computational real algebraic geometry.
The estimate of number of root invariant regions for polynomial parametric
families of polynomial and matrices is given. For the case of two parametric
family more sharp estimate is proven. Theoretic results are supported by
various numerical simulations that show higher precision of presented methods
with respect to traditional ones. The presented methods are inherently global
and could be applied for studying -decomposition for the space of parameters
as a whole instead of some prescribed regions. For symbolic computations the
Maple v.14 software and its package RegularChains are used.Comment: 16 pages, 8 figure
Hyperbolic Ax-Lindemann theorem in the cocompact case
We prove an analogue of the classical Ax-Lindemann theorem in the context of
compact Shimura varieties. Our work is motivated by J. Pila's strategy for
proving the Andr\'e-Oort conjecture unconditionallyComment: To appear in Duke Mathematical Journa
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
Tamagawa numbers of polarized algebraic varieties
Let be an ample metrized invertible sheaf on
a smooth quasi-projective algebraic variety defined over a number field.
Denote by the number of rational points in having -height . We consider the problem of a geometric and arithmetic
interpretation of the asymptotic for as in
connection with recent conjectures of Fujita concerning the Minimal Model
Program for polarized algebraic varieties.
We introduce the notions of -primitive varieties and -primitive fibrations. For -primitive varieties over we
propose a method to define an adelic Tamagawa number which
is a generalization of the Tamagawa number introduced by Peyre for
smooth Fano varieties. Our method allows us to construct Tamagawa numbers for
-Fano varieties with at worst canonical singularities. In a series of
examples of smooth polarized varieties and singular Fano varieties we show that
our Tamagawa numbers express the dependence of the asymptotic of on the choice of -adic metrics on .Comment: 54 pages, minor correction
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