349 research outputs found
On the divisibility of class numbers of quadratic fields and the solvability of Diophantine equations
In this paper we provide criteria for the insolvability of the Diophantine
equation . This result is then used to determine the class number of
the quadratic field . We also determine some criteria
for the divisibility of the class number of the quadratic field
and this result is then used to discuss the solvability
of the Diophantine equation
The cyclotomic Iwasawa main conjecture for Hilbert cuspforms with complex multiplication
We deduce the cyclotomic Iwasawa main conjecture for Hilbert modular
cuspforms with complex multiplication from the multivariable main conjecture
for CM number fields. To this end, we study in detail the behaviour of the
-adic -functions and the Selmer groups attached to CM number fields under
specialisation procedures.Comment: 83 pages, no figure
On the Iwasawa theory of CM fields for supersingular primes
The goal of this article is two-fold: First, to prove a (two-variable) main
conjecture for a CM field without assuming the -ordinary hypothesis of
Katz, making use of what we call the Rubin-Stark -restricted
Kolyvagin systems which is constructed out of the conjectural Rubin-Stark Euler
system of rank . (We are also able to obtain weaker unconditional results in
this direction.) Second objective is to prove the Park-Shahabi plus/minus main
conjecture for a CM elliptic curve defined over a general totally real
field again using (a twist of the) Rubin-Stark Kolyvagin system. This latter
result has consequences towards the Birch and Swinnerton-Dyer conjecture for
.Comment: 39 pages, to appear in Transactions of AMS (might slightly differ
from the final published version
On the computation of the class numbers of real Abelian fields
Siirretty Doriast
On The Relative Class Number of Special Cyclotomic Fields
Let p be an odd prime, p be a primitive pth root of unity and h��p be the relative class number of the pth cyclotomic fi eld Q( p) over the rationals Q defi ned by p. The main purpose of this paper is to discuss arithmetic properties of factors of h��p for an odd prime p of the form p = 4q + 1 with q a prime
On the Iwasawa main conjectures for modular forms at non-ordinary primes
In this paper, we prove under mild hypotheses the Iwasawa main conjectures of
Lei--Loeffler--Zerbes for modular forms of weight at non-ordinary primes.
Our proof is based on the study of the two-variable analogues of these
conjectures formulated by B\"uy\"ukboduk--Lei for imaginary quadratic fields in
which splits, and on anticyclotomic Iwasawa theory. As application of our
results, we deduce the -part of the Birch and Swinnerton-Dyer formula in
analytic ranks or for abelian varieties over of -type for non-ordinary primes
Differential principal factors and Polya property of pure metacyclic fields
Barrucand and Cohn's theory of principal factorizations in pure cubic fields
and their Galois closures
with types is generalized to pure
quintic fields and pure metacyclic fields
with possible types. The
classification is based on the Galois cohomology of the unit group ,
viewed as a module over the automorphism group of
over the cyclotomic field , by making use of theorems
by Hasse and Iwasawa on the Herbrand quotient of the unit norm index
by the number of
primitive ambiguous principal ideals, which can be interpreted as principal
factors of the different . The precise structure of the
group of differential principal factors is determined with the aid of kernels
of norm homomorphisms and central orthogonal idempotents. A connection with
integral representation theory is established via class number relations by
Parry and Walter involving the index of subfield units .
Generalizing criteria for the Polya property of Galois closures
of pure cubic fields
by Leriche and Zantema, we prove that pure
metacyclic fields of only type
cannot be Polya fields. All theoretical results are underpinned by extensive
numerical verifications of the possible types and their statistical
distribution in the range of normalized radicands.Comment: 30 pages, 10 sections, 6 table
An Equivariant Main Conjecture in Iwasawa Theory and Applications
We construct a new class of Iwasawa modules, which are the number field
analogues of the p-adic realizations of the Picard 1-motives constructed by
Deligne in the 1970s and studied extensively from a Galois module structure
point of view in our recent work. We prove that the new Iwasawa modules are of
projective dimension 1 over the appropriate profinite group rings. In the
abelian case, we prove an Equivariant Main Conjecture, identifying the first
Fitting ideal of the Iwasawa module in question over the appropriate profinite
group ring with the principal ideal generated by a certain equivariant p-adic
L-function. This is an integral, equivariant refinement of the classical Main
Conjecture over totally real number fields proved by Wiles in 1990. Finally, we
use these results and Iwasawa co-descent to prove refinements of the
(imprimitive) Brumer-Stark Conjecture and the Coates-Sinnott Conjecture, away
from their 2-primary components, in the most general number field setting. All
of the above is achieved under the assumption that the relevant prime p is odd
and that the appropriate classical Iwasawa mu-invariants vanish (as conjectured
by Iwasawa.)Comment: 52 page
On the -invariant of the cyclotomic derivative of Katz p-adic L-function
When the branch character has root number -1, the corresponding
anticyclotomic Katz p-adic L-function identically vanishes. In this case, we
study the -invariant of the cyclotomic derivative of Katz p-adic
L-function. As an application, this proves the non-vanishing of the
anticyclotomic regulator of a self-dual CM modular form with the root number
-1. The result also plays a crucial role in the recent work of Hsieh on the
Eisenstein ideal approach to a one-sided divisibility of the CM main
conjecture.Comment: 15 pages, revised, to appear in J. Institut Math. Jussie
Squarefree values of trinomial discriminants
The discriminant of a trinomial of the form has the form
if and are relatively prime. We
investigate when these discriminants have nontrivial square factors. We explain
various unlikely-seeming parametric families of square factors of these
discriminant values: for example, when is congruent to 2 (mod 6) we have
that always divides . In addition, we
discover many other square factors of these discriminants that do not fit into
these parametric families. The set of primes whose squares can divide these
sporadic values as varies seems to be independent of , and this set can
be seen as a generalization of the Wieferich primes, those primes such that
is congruent to 1 (mod ). We provide heuristics for the density
of squarefree values of these discriminants and the density of these "sporadic"
primes.Comment: 22 pages, 1 table. Minor revisions from version 1, including three
new reference
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