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Some new maps and ideals in classical Iwasawa theory with applications

David Solomon — 2014

Acta Arithmetica

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 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,...

On the classgroups of imaginary abelian fields

David Solomon — 1990

Annales de l'institut Fourier

Let p be an odd prime, χ an odd, p -adic Dirichlet character and K the cyclic imaginary extension of Q associated to χ . We define a “ χ -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 , χ - 1 . This uses the results of Mazur and Wiles in Iwasawa theory over Q . The more difficult case, in which p divides the order of χ is our chief concern. In this case the result is new and confirms an earlier conjecture of G....

p -adic Abelian Stark conjectures at s = 1

David Solomon — 2002

Annales de l’institut Fourier

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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