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 KK-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