Rubin’s Integral Refinement of the Abelian Stark Conjecture
نویسندگان
چکیده
This paper is a survey of results obtained by the present author towards proving Rubin’s integral version of Stark’s conjecture for abelian L– functions of arbitrary order of vanishing at the origin. Rubin’s conjecture is stated and its links to the classical integral Stark conjecture for L–functions of order of vanishing 1 are discussed. A weaker version of Rubin’s conjecture formulated by the author in [P4] is also stated and its links to Rubin’s conjecture are discussed. Evidence in support of the validity of Rubin’s conjecture is provided. A series of applications of Rubin’s conjecture to the theory of Euler Systems, groups of special units and Gras-type conjectures are given.
منابع مشابه
The Rubin–stark Conjecture for a Special Class of Function Field Extensions
We prove a strong form of the Brumer–Stark Conjecture and, as a consequence, a strong form of Rubin’s integral refinement of the abelian Stark Conjecture, for a large class of abelian extensions of an arbitrary characteristic p global field k. This class includes all the abelian extensions K/k contained in the compositum kp∞ := kp · k∞ of the maximal pro-p abelian extension kp/k and the maximal...
متن کاملOn a Refined Stark Conjecture for Function Fields
We prove that a refinement of Stark’s Conjecture formulated by Rubin in [14] 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.
متن کاملIntegral and p-adic Refinements of the Abelian Stark Conjecture
We give a formulation of the abelian case of Stark’s Main Conjecture in terms of determinants of projective modules and briefly show how this formulation leads naturally to its Equivariant Tamagawa Number Conjecture (ETNC) – type integral refinements. We discuss the Rubin-Stark integral refinement of an idempotent p1 iece of Stark’s Abelian Main Conjecture. In the process, we give a new formula...
متن کاملOn the Rubin–stark Conjecture for a Special Class of Cm Extensions of Totally Real Number Fields
We use Greither’s recent results on Brumer’s Conjecture to prove Rubin’s integral version of Stark’s Conjecture, up to a power of 2, for an infinite class of CM extensions of totally real number fields, called “nice extensions”. As a consequence, we show that the Brumer–Stark Conjecture is true for “nice extensions”, up to a power of 2.
متن کاملSpecial Values of Abelian L - Functions at S = 0
In [12], Stark formulated his far-reaching refined conjecture on the first derivative of abelian (imprimitive) L–functions of order of vanishing r = 1 at s = 0. In [10], Rubin extended Stark’s refined conjecture to describe the r-th derivative of abelian (imprimitive) L-functions of order of vanishing r at s = 0, for arbitrary values r. However, in both Stark’s and Rubin’s setups, the order of ...
متن کامل