635 research outputs found

### On a Refined Stark Conjecture for Function Fields

We prove that a refinement of Stark's Conjecture formulated by Rubin is true
up to primes dividing the order of the Galois group, for finite, abelian
extensions of function fields over finite fields. We also show that in the case
of constant field extensions a statement stronger than Rubin's holds true

### Hecke characters and the $K$-theory of totally real and CM number fields

Let $F/K$ be an abelian extension of number fields with $F$ either CM or
totally real and $K$ totally real. If $F$ is CM and the Brumer-Stark conjecture
holds for $F/K$, we construct a family of $G(F/K)$--equivariant Hecke
characters for $F$ with infinite type equal to a special value of certain
$G(F/K)$--equivariant $L$-functions. Using results of Greither-Popescu on the
Brumer-Stark conjecture we construct $l$-adic imprimitive versions of these
characters, for primes $l> 2$. Further, the special values of these $l$-adic
Hecke characters are used to construct $G(F/K)$-equivariant
Stickelberger-splitting maps in the $l$-primary Quillen localization sequence
for $F$, extending the results obtained in 1990 by Banaszak for $K = \Bbb Q$.
We also apply the Stickelberger-splitting maps to construct special elements in
the $l$-primary piece $K_{2n}(F)_l$ of $K_{2n}(F)$ and analyze the Galois
module structure of the group $D(n)_l$ of divisible elements in $K_{2n}(F)_l$,
for all $n>0$. If $n$ is odd and coprime to $l$ and $F = K$ is a fairly general
totally real number field, we study the cyclicity of $D(n)_l$ in relation to
the classical conjecture of Iwasawa on class groups of cyclotomic fields and
its potential generalization to a wider class of number fields. Finally, if $F$
is CM, special values of our $l$-adic Hecke characters are used to construct
Euler systems in the odd $K$-groups with coefficients $K_{2n+1}(F, \Bbb
Z/l^k)$, for all $n>0$. These are vast generalizations of Kolyvagin's Euler
system of Gauss sums and of the $K$-theoretic Euler systems constructed in
Banaszak-Gajda when $K = \Bbb Q$.Comment: 38 page

### 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

### An Equivariant Tamagawa Number Formula for Drinfeld Modules and Applications

We fix data $(K/F, E)$ consisting of a Galois extension $K/F$ of
characteristic $p$ global fields with arbitrary abelian Galois group $G$ and a
Drinfeld module $E$ defined over a certain Dedekind subring of $F$. For this
data, we define a $G$-equivariant $L$-function $\Theta_{K/F}^E$ and prove an
equivariant Tamagawa number formula for certain Euler-completed versions of its
special value $\Theta_{K/F}^E(0)$. This generalizes Taelman's class number
formula for the value $\zeta_F^E(0)$ of the Goss zeta function $\zeta_F^E$
associated to the pair $(F, E)$. Taelman's result is obtained from our result
by setting $K=F$. As a consequence, we prove a perfect Drinfeld module analogue
of the classical (number field) refined Brumer--Stark conjecture, relating a
certain $G$-Fitting ideal of Taelman's class group $H(E/K)$ to the special
value $\Theta_{K/F}^E(0)$ in question

### An unconditional main conjecture in Iwasawa theory and applications

We improve upon the recent keystone result of Dasgupta-Kakde on the $\Bbb
Z[G(H/F)]^-$-Fitting ideals of certain Selmer modules $Sel_S^T(H)^-$ associated
to an abelian, CM extension $H/F$ of a totally real number field $F$ and use
this to compute the $\Bbb Z_p[[G(H_\infty/F)]]^-$-Fitting ideal of the Iwasawa
module analogues $Sel_S^T(H_\infty)_p^-$ of these Selmer modules, where
$H_\infty$ is the cyclotomic $\Bbb Z_p$-extension of $H$, for an odd prime $p$.
Our main Iwasawa theoretic result states that the $\Bbb
Z_p[[G(H_\infty/F]]^-$-module $Sel_S^T(H_\infty)_p^-$ is of projective
dimension $1$, is quadratically presented, and that its Fitting ideal is
principal, generated by an equivariant $p$-adic $L$-function
$\Theta_S^T(H_\infty/F)$. Further, we establish a perfect duality pairing
between $Sel_S^T(H_\infty)_p^-$ and a certain $\Bbb
Z_p[[G(H_\infty/F)]]^-$-module $\mathcal M_S^T(H_\infty)^-$, essentially
introduced earlier by Greither-Popescu. As a consequence, we recover the
Equivariant Main Conjecture for the Tate module $T_p(\mathcal
M_S^T(H_\infty))^-$, proved by Greither-Popescu under the hypothesis that the
classical Iwasawa $\mu$-invariant associated to $H$ and $p$ vanishes. As a
further consequence, we give an unconditional proof of the refined
Coates-Sinnott Conjecture, proved by Greither-Popescu under the same $\mu=0$
hypothesis, and also recently proved unconditionally but with different methods
by Johnston-Nickel, regarding the $\Bbb Z[G(H/F)]$-Fitting ideals of the higher
Quillen $K$-groups $K_{2n-2}(\mathcal O_{H,S})$, for all $n\geq 2$. Finally, we
combine the techniques developed in the process with the method of
''Taylor-Wiles primes'' to strengthen further the keystone result of
Dasgupta-Kakde and prove, as a consequence, a conjecture of Burns-Kurihara-Sano
on Fitting ideals of Selmer groups of CM number fields

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