By Dr. Serge Lang (auth.)

Kummer's paintings on cyclotomic fields lead the way for the advance of algebraic quantity conception mostly via Dedekind, Weber, Hensel, Hilbert, Takagi, Artin and others. even if, the luck of this common conception has tended to imprecise targeted evidence proved by means of Kummer approximately cyclotomic fields which lie deeper than the final concept. For a protracted interval within the twentieth century this element of Kummer's paintings turns out to were principally forgotten, apart from a couple of papers, between that are these via Pollaczek [Po], Artin-Hasse [A-H] and Vandiver [Va]. within the mid 1950's, the idea of cyclotomic fields used to be taken up back by means of Iwasawa and Leopoldt. Iwasawa considered cyclotomic fields as being analogues for quantity fields of the consistent box extensions of algebraic geometry, and wrote an excellent series of papers investigating towers of cyclotomic fields, and extra more often than not, Galois extensions of quantity fields whose Galois team is isomorphic to the additive staff of p-adic integers. Leopoldt targeting a hard and fast cyclotomic box, and tested numerous p-adic analogues of the classical advanced analytic classification quantity formulation. particularly, this led him to introduce, with Kubota, p-adic analogues of the complicated L-functions hooked up to cyclotomic extensions of the rationals. eventually, within the past due 1960's, Iwasawa [Iw 1 I] . made the basic discovery that there has been an in depth connection among his paintings on towers of cyclotomic fields and those p-adic L-functions of Leopoldt-Kubota.

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R() n R = (R()' n R)-. Lemma 2. Proof Let T = R()' n R. Clearly T- c e- R() = R() and T- c R, so the inclusion ~ is obvious. Conversely, let IY. E R() n R. It will suffice to prove that IY. E R()' (because IY. E Rand IY. -). Write From the hypothesis that IY. )2 == 0 (mod Z) for all e prime to m, so that - L z(b)b == -21 L z(b) (mod Z). 1 m b b We contend that L z(b)b == 0 (mod m) and L z(b) == 0 (mod 2). This is obvious if m is odd. Suppose m even, so m is divisible by 4. Write m = 4mo. Each b is odd, and L z(b)b == 0 (mod 2mo) so 2: z(b) is even.

The zeta function Z(V, T) is defined by the conditions Z'IZ(T) =- LNT v v -1 and Z(O) = 1. b T ) Z(V(d), T) - (1 _ T)(1 - qT) This is best seen by taking the logarithmic derivative of the last expression on the right-hand side. The operator Jf-+ I'IJ is a homomorphism, so we take the operator for each linear term. Inverting a geometric series we see that the logarithmic derivative of the last expression on the right-hand side has precisely the power series Since it has the value 1 at T properties.

The Iwasawa-Leopoldt conjecture predicts an isomorphism where C:- is the p-primary part of the (-I)-eigenspace of the ideal class group in Q(flp). On the other hand, Kubert-Lang [KL 7] establish an isomorphism where 'i&'O(Xl(P)) is the cuspidal divisor class group on the modular curve X1(p), generated by the cusps lying above the relational cusp on Xo(p), Consequently, we expect a commutative diagram: It remains a problem to give a direct isomorphism at the bottom, from some sort of geometric construction.