Iwasawa Main Conjecture for the Carlitz cyclotomic extension and applications

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We prove an Iwasawa Main Conjecture for the class group of the p-cyclotomic extension F of the function field Fq(θ) (p is a prime of Fq[θ]), showing that its Fitting ideal is generated by a Stickelberger element. We use this and a link between the Stickelberger element and a p-adic L-function to prove a close analog of the Ferrero-Washington theorem for F and to provide informations on the p-adic valuations of the Bernoulli-Goss numbers β(j) (i.e., on the values of the Goss ζ-function at negative integers).