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A remark on arithmetic equivalence and the normset

Jim Coykendall (2000)

Acta Arithmetica

1. Introduction. Number fields with the same zeta function are said to be arithmetically equivalent. Arithmetically equivalent fields share much of the same properties; for example, they have the same degrees, discriminants, number of both real and complex valuations, and prime decomposition laws (over ℚ). They also have isomorphic unit groups and determine the same normal closure over ℚ [6]. Strangely enough, it has been shown (for example [4], or more recently [6] and [7]) that this does...

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 the existence...

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