By Dr. Serge Lang (auth.)

Kummer's paintings on cyclotomic fields cleared the path for the improvement of algebraic quantity conception mostly by means of Dedekind, Weber, Hensel, Hilbert, Takagi, Artin and others. besides the fact that, the luck of this basic thought has tended to imprecise exact proof proved via Kummer approximately cyclotomic fields which lie deeper than the final thought. For an extended interval within the twentieth century this point of Kummer's paintings turns out to were principally forgotten, with the exception of a couple of papers, between that are these by way of Pollaczek [Po], Artin-Hasse [A-H] and Vandiver [Va]. within the mid 1950's, the idea of cyclotomic fields was once taken up back via Iwasawa and Leopoldt. Iwasawa considered cyclotomic fields as being analogues for quantity fields of the consistent box extensions of algebraic geometry, and wrote a superb series of papers investigating towers of cyclotomic fields, and extra typically, Galois extensions of quantity fields whose Galois crew is isomorphic to the additive workforce of p-adic integers. Leopoldt focused on a set cyclotomic box, and verified a number of p-adic analogues of the classical advanced analytic classification quantity formulation. specifically, this led him to introduce, with Kubota, p-adic analogues of the advanced L-functions connected to cyclotomic extensions of the rationals. ultimately, 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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**Example text**

It is obvious that We shall determine the extent to which J =F I in §2. We have: degI(k)(N) = pt, where t is the maximum integer such that k == 0 mod cp(pt). This is obvious, because deg I(k)(N) is generated by the integers 1 - ck with c prime to p. 1. (i) We-have RlJ~ n R = I(k)lJ~. In jact, if an element, E R is such that ,lJ' E R, then, E I(k). (ii) On the other hand, letting I~k) = ZP/(k), we have If an element, E Rp is such that ,lJ E Rp then, E I~k). Proof First we prove that for any prime ~ 2, we have IlJ' c R, and IplJ c Rp.

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. Then L z(b)b == I L z(b)(mod mZ), thus proving also the first congruence. Only the second will be used.

Suppose N = p is prime '# 2, 3. 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.