32,801 research outputs found
p-adic equidistribution of CM points
Let be a modular curve and consider a sequence of Galois orbits of CM
points in , whose -conductors tend to infinity. Its equidistribution
properties in and in the reductions of modulo primes different
from are well understood. We study the equidistribution problem in the
Berkovich analytification of .
We partition the set of CM points of sufficiently high conductor in into finitely many explicit \emph{basins} , indexed by the
irreducible components of the mod- reduction of the canonical model of
. We prove that a sequence of local Galois orbits of CM points with
-conductor going to infinity has a limit in if and only if
it is eventually supported in a single basin . If so, the limit is the
unique point of whose mod- reduction is the generic point
of .
The result is proved in the more general setting of Shimura curves over
totally real fields. The proof combines Gross's theory of quasicanonical
liftings with a new formula for the intersection numbers of CM curves and
vertical components in a Lubin--Tate space.Comment: Some improvements in the exposition. 23 pages, 1 new figur
Birationally superrigid cyclic triple spaces
We prove the birational superrigidity and the nonrationality of a cyclic
triple cover of branched over a nodal hypersurface of degree
for . In particular, the obtained result solves the problem of the
birational superrigidity of smooth cyclic triple spaces. We also consider
certain relevant problems.Comment: 43 page
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