Let p be a prime number and n a non-negative integer. We denote by h p,n the class number of the n-th layer of the cyclotomic Z p -extension of Q. Let l be a prime number. In this paper, we assume that p is odd and consider the l-divisibility of h p,n . Let f be the inertia degree of l in the p-th cyclotomic field and s the maximal exponent such that p s divides l p−1 − 1. Set r = min{n, s}. We define a certain explicit constant G 1 (p, r, f ) in terms of the property of the residue class of l modulo p r . If l is larger than G 1 (p, r, f ), then the integer h p,n /h p,n−1 is coprime with l. Our proof refines Horie's method.
Introduction.Let p be a prime number and μ m the group of all m-th roots of unity in C and putwhose Galois group Gal(B p,∞ /Q) is topologically isomorphic to the p-adic integer ring Z p as additive groups. Let B p,n be the unique subfield of B p,∞ which is cyclic of degree p n over Q and h p,n its class number. In the case p = 2, Weber [26] showed that 2 does not divide h 2,n for any positive integer n and he also showed h 2,1 = h 2,2 = h 2,3 = 1. Based on these results, Weber asked whether h 2,n = 1 for any positive integer n. Then we consider a generalized version of his problem:
WEBER'S CLASS NUMBER PROBLEM. Is the class number h p,n equal to one for any positive integer n?This problem has been studied by Bauer [1], Cohn [2], Masley [19], who showed h 2,4 = 1. Later, van der Linden [17] showed h 2,5 = 1 or 97. However, Komatsu and Fukuda [4] showed that 97 does not divide h 2,n for any positive integer n. Hence we have h 2,5 = 1. In [1] and [17], we know that h p,n = 1 for (p, n) ∈ {(3, 1), (3, 2), (3, 3), (5, 1), (7, 1)}. Linden also showed that h p,n = 1 for (p, n) ∈ {(2, 6), (3, 4), (5, 2), (11, 1), (13, 1)} under the generalized Riemann hypothesis.However, the direct calculation of h p,n is extremely difficult for large p n . Therefore, in order to break the wall of the computational complexity, we study the l-divisibility of h p,n for a prime number l and for all positive integer n: PROBLEM. Does a prime number l divide h p,n for any positive integer n?