27 research outputs found
Poitou-Tate without restrictions on the order
The Poitou-Tate sequence relates Galois cohomology with restricted
ramification of a finite Galois module over a global field to that of the
dual module under the assumption that is a unit away from the allowed
ramification set. We remove the assumption on by proving a generalization
that allows arbitrary "ramification sets" that contain the archimedean places.
We also prove that restricted products of local cohomologies that appear in the
Poitou-Tate sequence may be identified with derived functor cohomology of an
adele ring. In our proof of the generalized sequence we adopt this derived
functor point of view and exploit properties of a natural topology carried by
cohomology of the adeles.Comment: 28 pages; final version, to appear in Mathematical Research Letter
Topology on cohomology of local fields
Arithmetic duality theorems over a local field are delicate to prove if
. In this case, the proofs often exploit topologies
carried by the cohomology groups for commutative finite type
-group schemes . These "\v{C}ech topologies", defined using \v{C}ech
cohomology, are impractical due to the lack of proofs of their basic
properties, such as continuity of connecting maps in long exact sequences. We
propose another way to topologize : in the key case ,
identify with the set of isomorphism classes of objects of the
groupoid of -points of the classifying stack and invoke
Moret-Bailly's general method of topologizing -points of locally of finite
type -algebraic stacks. Geometric arguments prove that these "classifying
stack topologies" enjoy the properties expected from the \v{C}ech topologies.
With this as the key input, we prove that the \v{C}ech and the classifying
stack topologies actually agree. The expected properties of the \v{C}ech
topologies follow, which streamlines a number of arithmetic duality proofs
given elsewhere.Comment: 36 pages; final version, to appear in Forum of Mathematics, Sigm
The Manin constant in the semistable case
For an optimal modular parametrization of an
elliptic curve over of conductor , Manin conjectured the
agreement of two natural -lattices in the -vector space
. Multiple authors generalized his conjecture to higher
dimensional newform quotients. We prove the Manin conjecture for semistable
, give counterexamples to all the proposed generalizations, and prove
several semistable special cases of these generalizations. The proofs establish
general relations between the integral -adic etale and de Rham cohomologies
of abelian varieties over -adic fields and exhibit a new exactness result
for Neron models.Comment: 28 pages; final version, to appear in Compositio Mathematic
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
A modular description of
As we explain, when a positive integer is not squarefree, even over
the moduli stack that parametrizes generalized elliptic curves
equipped with an ample cyclic subgroup of order does not agree at the cusps
with the -level modular stack defined by
Deligne and Rapoport via normalization. Following a suggestion of Deligne, we
present a refined moduli stack of ample cyclic subgroups of order that does
recover over for all . The resulting modular
description enables us to extend the regularity theorem of Katz and Mazur:
is also regular at the cusps. We also prove such regularity
for and several other modular stacks, some of which have
been treated by Conrad by a different method. For the proofs we introduce a
tower of compactifications of the stack that
parametrizes elliptic curves---the ability to vary in the tower permits
robust reductions of the analysis of Drinfeld level structures on generalized
elliptic curves to elliptic curve cases via congruences.Comment: 67 pages; final version, to appear in Algebra and Number Theor