48 research outputs found
Note on the rational points of a pfaff curve
Let X be the graph in the plane of a pfaffian function f (in the sense of
Khovanskii). Suppose X is not algebraic. This note gives an upper bound for the
number of rational points on X of height up to X. The bound is uniform in the
order and degre of f.Comment: 8 page
Counting points on curves over families in polynomial time
This note concerns the theoretical algorithmic problem of counting rational
points on curves over finite fields. It explicates how the algorithmic scheme
introduced by Schoof and generalized by the author yields an algorithm whose
running time is uniformly polynomial time for curves in families.Comment: 7 page
The Andre-Oort conjecture for the moduli space of Abelian Surfaces
We provide an unconditional proof of the Andr\'e-Oort conjecture for the
coarse moduli space of principally polarized Abelian
surfaces, following the strategy outlined by Pila-Zannier.Comment: 14 page
Rational points in periodic analytic sets and the Manin-Mumford conjecture
We present a new proof of the Manin-Mumford conjecture about torsion points
on algebraic subvarieties of abelian varieties. Our principle, which admits
other applications, is to view torsion points as rational points on a complex
torus and then compare (i) upper bounds for the number of rational points on a
transcendental analytic variety (Bombieri-Pila-Wilkie) and (ii) lower bounds
for the degree of a torsion point (Masser), after taking conjugates. In order
to be able to deal with (i), we discuss (Thm. 2.1) the semi-algebraic curves
contained in an analytic variety supposed invariant for translations by a full
lattice, which is a topic with some independent motivation.Comment: 12 page
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
Ax-Schanuel for Shimura varieties
We prove the Ax-Schanuel theorem for a general (pure) Shimura variety