RÉSUMÉ

For an imaginary abelian field, I define the Stickelberger ideal by a slightly different method from Sinnott's, and propose a conjecture that the Fitting ideal of the minus ideal class group is equal to the Stickelberger ideal. This conjecture gives more information on the structure of ideal class groups than the usual Iwasawa main conjecture which is formulated by using characteristic ideals over cyclotomic Z_{p}-extensions. I give several partial results on this conjecture, and some application. I will also propose and discuss an analogous conjecture for the Selmer group of an elliptic curve.

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