We shall study some explicit connections between (1) the Vandiver conjecture on the class number of the real cyclotomic field Q(cos(2π/l)) and (2) the images of various Galois representations induced from the power series representation (constructed and studied by Ihara, Anderson, Coleman, etc.) of Gal(Q/Q(μI∞)) which describes universally the Galois action on the Fermat curves of l-power degrees. One such connection was first discovered by Coleman. In the case of the original power series representation, we shall also describe the difference between the "expected image" and the actual Galois image in terms of a certain invariant of Iwasawa type.
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