### On a construction on p-units in abelian fields.

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We introduce a new ideal of the p-adic Galois group-ring associated to a real abelian field and a related ideal for imaginary abelian fields, Both result from an equivariant, Kummer-type pairing applied to Stark units in a ${\mathbb{Z}}_{p}$-tower of abelian fields, and is linked by explicit reciprocity to a third ideal studied more generally in [D. Solomon, Acta Arith. 143 (2010)]. This leads to a new and unifying framework for the Iwasawa theory of such fields including a real analogue of Stickelberger’s Theorem,...

Let $p$ be an odd prime, $\chi $ an odd, $p$-adic Dirichlet character and $K$ the cyclic imaginary extension of $\mathbf{Q}$ associated to $\chi $. We define a “$\chi $-part” of the Sylow $p$-subgroup of the class group of $K$ and prove a result relating its $p$-divisibility to that of the generalized Bernoulli number ${B}_{1,{\chi}^{-1}}$. This uses the results of Mazur and Wiles in Iwasawa theory over $\mathbf{Q}$. The more difficult case, in which $p$ divides the order of $\chi $ is our chief concern. In this case the result is new and confirms an earlier conjecture of G....

A $p$-adic version of Stark’s Conjecture at $s=1$ is attributed to J.-P. Serre and stated (faultily) in Tate’s book on the Conjecture. Building instead on our previous paper (and work of Rubin) on the complex abelian case, we give a new approach to such a conjecture for real ray-class extensions of totally real number fields. We study the coherence of our $p$-adic conjecture and then formulate some integral refinements, both alone and in combination with its complex analogue. A ‘Weak Combined Refined’ version...

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