Compositio Mathematica

Stickelberger Elements for Cyclic Extensions and the Order of Vanishing of Abelian L-Functions at s=0

Joongul Lee a1
a1 School of Mathematics, Korea Institute for Advanced Study, 207-43 Cheongryangri-dong, Dongdaemun-gu, Seoul, Republic of Korea

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We study the Stickelberger element of a cyclic extension of global fields of prime power degree. Assuming that S contains an almost splitting place, we show that the Stickelberger element is contained in a power of the relative augmentation ideal whose exponent is at least as large as Gross's prediction. This generalizes the work of Tate (see Section 4) on a refinement of Gross's conjecture in the cyclic case. We also present an example for which Tate's prediction does not hold.

Key Words: class numbers; Gross's conjecture; special values of Abelian L-functions; Stickelberger elements.