Mahler measure and Weber's class number problem in the cyclotomic $\boldsymbol{Z}_p$-extension of $\boldsymbol{Q}$ for odd prime number $p$

Abstract

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?

Triviality of the ℓ-class groups in -extensions of for split primes p ≡ 1 modulo 4

“…Much progress has been made in improving the bounds arising from Horie's arguments, particularly in the work of Morisawa and Okazaki [20].…”

Tate–Shafarevich groups in the cyclotomic Zˆ-extension and Weber's class number problem

“…Indeed, one may ask if the arithmetic of these fields is as smooth as it is conjectured (for the class group C Q(N ) ) by many authors after many verifications and partial proofs [2,5,10,11,12,13,14,15,34,35,36,37,38,39,40,46,47,48,49,50,51,52,59]. The triviality of C Q(ℓ n ) has, so far, no counterexamples as ℓ, n, p vary, but that of the Tate-Shafarevich group T Q(ℓ n ) (or more generally T Q(N ) ) is, on the contrary, not true as we shall see numerically, and, for composite N , few C Q(N ) = 1 have been discovered.…”

On the class number of a real abelian field of prime conductor

“…, respectively. By ( 5), the assumptions of the proposition and theorem of [14] are satisfied in our setting. From the above, we obtain M ( ) ≥ ω r /2 .…”

“…One was used in Horie's paper to prove [5,Theorem 3]. The other is the Mahler measure, which played a role in Morisawa and Okazaki's paper [14] in sharpening one of Horie's results.…”

“…where ι runs over all embeddings of F into R. The measure * is effectively used in [5,6], and the Mahler measure M ( * ) is used in [14].…”