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A Stark conjecture “over 𝐙 ” for abelian L -functions with multiple zeros

Karl Rubin — 1996

Annales de l'institut Fourier

Suppose K / k is an abelian extension of number fields. Stark’s conjecture predicts, under suitable hypotheses, the existence of a global unit ϵ of K such that the special values L ' ( χ , 0 ) for all characters χ of Gal / ( K / k ) can be expressed as simple linear combinations of the logarithms of the different absolute values of ϵ . In this paper we formulate an extension of this conjecture, to attempt to understand the values L ( r ) ( χ , 0 ) when the order of vanishing r may be greater than one. This conjecture no longer predicts...

Anticyclotomic Iwasawa theory of CM elliptic curves

Adebisi AgboolaBenjamin Howard — 2006

Annales de l’institut Fourier

We study the Iwasawa theory of a CM elliptic curve E in the anticyclotomic Z p -extension of the CM field, where p is a prime of good, ordinary reduction for E . When the complex L -function of E vanishes to even order, Rubin’s proof of the two variable main conjecture of Iwasawa theory implies that the Pontryagin dual of the p -power Selmer group over the anticyclotomic extension is a torsion Iwasawa module. When the order of vanishing is odd, work of Greenberg show that it is not a torsion module. In...

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