325 research outputs found
Mordell-Lang in positive characteristic
We give a new proof of the Mordell-Lang conjecture in positive characteristic
for finitely generated subgroups. We also make some progress towards the full
Mordell-Lang conjecture in positive characteristic
The zero section of the universal semiabelian variety, and the double ramification cycle
We study the Chow ring of the boundary of the partial compactification of the
universal family of principally polarized abelian varieties (ppav). We describe
the subring generated by divisor classes, and compute the class of the partial
compactification of the universal zero section, which turns out to lie in this
subring. Our formula extends the results for the zero section of the universal
uncompactified family.
The partial compactification of the universal family of ppav can be thought
of as the first two boundary strata in any toroidal compactification of the
moduli space of ppav. Our formula provides a first step in a program to
understand the Chow groups of toroidal compactifications of the moduli of ppav,
especially of the perfect cone compactification, by induction on genus. By
restricting to the locus of Jacobians of curves, our results extend the results
of Hain on the double ramification (two-branch-point) cycle.Comment: Section 6, dealing with the Eliashberg problem for moduli of curves,
rewritten. A discussion of the extension of the Abel-Jacobi map added, the
resulting formula corrected. Final version, to appear in Duke Math.
The Yoneda isomorphism commutes with homology
We show that, for a right exact functor from an abelian category to abelian
groups, Yoneda's isomorphism commutes with homology and, hence, with functor
derivation. Then we extend this result to semiabelian domains. An
interpretation in terms of satellites and higher central extensions follows. As
an application, we develop semiabelian (higher) torsion theories and the
associated theory of (higher) universal (central) extensions.Comment: Fixed an inaccuracy in (3.6
On Mumford's construction of degenerating abelian varieties
We prove that a 1-dimnl family of abelian varieties with an ample sheaf
defining principal polarization can be canonically compactified (after a finite
base change) to a projective family with an ample sheaf. We show that the
central fiber (P,L), which we call an SQAV, has a canonical Cartier theta
divisor. We give a combinatorial definition for SQAVs and describe their
geometrical properties, in particular compute cohomologies of L^n, n\ge0.Comment: Final version, to appear in Tohoku Math.
Selmer groups as flat cohomology groups
Given a prime number , Bloch and Kato showed how the -Selmer
group of an abelian variety over a number field is determined by the
-adic Tate module. In general, the -Selmer group
need not be determined by the mod Galois representation ; we
show, however, that this is the case if is large enough. More precisely, we
exhibit a finite explicit set of rational primes depending on and
, such that is determined by for all . In the course of the argument we describe the flat cohomology
group of the ring of integers of
with coefficients in the -torsion of the N\'{e}ron
model of by local conditions for , compare them with the
local conditions defining , and prove that
itself is determined by for such . Our method
sharpens the known relationship between and
and continues to work for other
isogenies between abelian varieties over global fields provided that
is constrained appropriately. To illustrate it, we exhibit
resulting explicit rank predictions for the elliptic curve over certain
families of number fields.Comment: 22 pages; final version, to appear in Journal of the Ramanujan
Mathematical Societ
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