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(2006 / vol.9 / no.1)
Survey of Gross's conjecture and its refinements
Noboru Aoki
Pages. 31-34     



Let $K/k$ be an abelian extension of global fields (i.e. number fields or function fields of curves defined over finite field) with Galois group
$G$. In cite{gross}, B. Gross has conjectured a congruence
relation which relates the Stickelberger element in $G$ with the class number of
$k$ and the generalized regulator. The relation may be viewed as a
generalization of the classical class number formula which describes
the leading term of the Taylor expansion of $zeta_{k}(s)$ at $s=0$ in
terms of the class number and the regulator of $k$. This conjecture has been verified to be true in many important special cases but yet it remains to be proved in general.

Tate has discovered that both sides of the conjecture vanishes under mild restriction when $G$ is a cyclic group of prime power order, and he conjectured that a finer version of the congruence relation would hold. This idea has been developed independently by Burns cite{burns} and Aoki, Lee and Tan cite{alt} to cover the case when $G$ is an arbitrary finite abelian group with no restriction.

In this paper, we review the various conjectures mentioned above and discuss their relationships.



1. Gross's conjecture 2.Tate's refinement 3.Generalization of Tate's refinement



Stickelberger element, Abelian $L$-functions, class number



11R42