1,533 research outputs found
A note on Chudnovsky's Fuchsian equations
We show that four exceptional Fuchsian equations, each determined by the four
parabolic singularities, known as the Chudnovsky equations, are transformed
into each other by algebraic transformations. We describe equivalence of these
equations and their counterparts on tori. The latter are the Fuchsian equations
on elliptic curves and their equivalence is characterized by transcendental
transformations which are represented explicitly in terms of elliptic and theta
functions.Comment: Final version; LaTeX, 27 pages, 1 table, no figure
Hilbert modular surfaces for square discriminants and elliptic subfields of genus 2 function fields
We compute explicit rational models for some Hilbert modular surfaces
corresponding to square discriminants, by connecting them to moduli spaces of
elliptic K3 surfaces. Since they parametrize decomposable principally polarized
abelian surfaces, they are also moduli spaces for genus-2 curves covering
elliptic curves via a map of fixed degree. We thereby extend classical work of
Jacobi, Hermite, Bolza etc., and more recent work of Kuhn, Frey, Kani, Shaska,
V\"olklein, Magaard and others, producing explicit families of reducible
Jacobians. In particular, we produce a birational model for the moduli space of
pairs (C,E) of a genus 2 curve C and elliptic curve E with a map of degree n
from C to E, as well as a tautological family over the base, for 2 <= n <= 11.
We also analyze the resulting models from the point of view of arithmetic
geometry, and produce several interesting curves on them.Comment: 36 pages. Final versio
The sixth Painleve transcendent and uniformization of algebraic curves
We exhibit a remarkable connection between sixth equation of Painleve list
and infinite families of explicitly uniformizable algebraic curves. Fuchsian
equations, congruences for group transformations, differential calculus of
functions and differentials on corresponding Riemann surfaces, Abelian
integrals, analytic connections (generalizations of Chazy's equations), and
other attributes of uniformization can be obtained for these curves. As
byproducts of the theory, we establish relations between Picard-Hitchin's
curves, hyperelliptic curves, punctured tori, Heun's equations, and the famous
differential equation which Apery used to prove the irrationality of Riemann's
zeta(3).Comment: Final version. Numerous improvements; English, 49 pages, 1 table, no
figures, LaTe
Density of rational points on del Pezzo surfaces of degree one
We state conditions under which the set S(k) of k-rational points on a del
Pezzo surface S of degree 1 over an infinite field k of characteristic not
equal to 2 or 3 is Zariski dense. For example, it suffices to require that the
elliptic fibration over the projective line induced by the anticanonical map
has a nodal fiber over a k-rational point. It also suffices to require the
existence of a point in S(k) that does not lie on six exceptional curves of S
and that has order 3 on its fiber of the elliptic fibration. This allows us to
show that within a parameter space for del Pezzo surfaces of degree 1 over the
field of real numbers, the set of surfaces S defined over the field Q of
rational numbers for which the set S(Q) is Zariski dense, is dense with respect
to the real analytic topology. We also include conditions that may be satisfied
for every del Pezzo surface S and that can be verified with a finite
computation for any del Pezzo surface S that does satisfy them.Comment: 31 pages; the main results have not changed; the presentation has
been improved; a magma file that checks all computations may be obtained from
arXiv by downloading the source of this articl
Nonpersistence of resonant caustics in perturbed elliptic billiards
Caustics are curves with the property that a billiard trajectory, once
tangent to it, stays tangent after every reflection at the boundary of the
billiard table. When the billiard table is an ellipse, any nonsingular billiard
trajectory has a caustic, which can be either a confocal ellipse or a confocal
hyperbola. Resonant caustics ---the ones whose tangent trajectories are closed
polygons--- are destroyed under generic perturbations of the billiard table. We
prove that none of the resonant elliptical caustics persists under a large
class of explicit perturbations of the original ellipse. This result follows
from a standard Melnikov argument and the analysis of the complex singularities
of certain elliptic functions.Comment: 14 pages, 3 figure
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