Mahler measure of the Horie unit and Weber's class number problem in the cyclotomic Z<sub>3</sub>-extension of Q

Morisawa, Takayuki

doi:10.4064/aa153-1-3

Cited by 7 publications

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“…) is non-trivial for n large enough, then it is divisible by p ρn due to the Galois action on a non-trivial p-class of C * Q(ℓ n ) , which becomes oversized; see § 2.4 for more details showing that C * Q(ℓ n ) = 1 for n ≫ 0 does exist for any prime p ≥ 2 from a non-trivial result of Washington [56] and explicit deep analytic computations in [4,9,10,13,34,35,36,37,45,46,48,49] (e.g., [13,Corollary 1]). 2 2.3.…”

Section: 1mentioning

confidence: 99%

“…Class groups and torsion groups of abelian p-ramification, in Q(ℓ ∞ ). The invariants C Q(ℓ n ) and T Q(ℓ n ) , for all p = ℓ, are the fundamental invariants of Q(ℓ n ) and one may ask if the arithmetic of Q(ℓ n ) is as smooth as it is conjectured (for the class group) by many authors after many verifications and partial proofs [4,10,11,12,13,32,33,34,35,36,37,43,44,45,46,47,48,49]. The triviality of C Q(ℓ n ) has no counterexamples as ℓ, n, p vary, but that of T Q(ℓ n ) is, on the contrary, not true as we shall see numerically.…”

Section: Introductionmentioning

confidence: 99%

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Weber's class number problem and $p$-rationality in the cyclotomic $\widehat{\mathbb{Z}}$-extension of $\mathbb{Q}$

Gras

2020

Preprint

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Let K := Q(ℓ n ) be the nth layer of the cyclotomic Z ℓ -extension. It is conjectured that K is principal (Weber's conjecture for ℓ = 2). Many studies (Ichimura-Miller-Morisawa-Nakajima-Okazaki) go in this direction. Nevertheless, we examine if a counterexample may be possible. For this, computations show that the p-torsion group T K of the Galois group of the maximal abelian p-ramified pro-p-extension of K is not always trivial; whence the relevance of the conjecture since, where C K is the p-class group, and R K the normalized p-adic regulator. We give a new method (Theorem 4.6 testing # T K = 1), allowing larger values of ℓ n than those of the literature. Finally, we search in the cyclotomic Z-extension, cases of non-trivial class groups using genus theory related to a deep property of R K (Theorem 6.3); we only find again the three known cases (Fukuda-Komatsu-Horie). Contents 1. Introduction 1.1. Class groups and torsion groups of abelian p-ramification, in Q(ℓ ∞ ) 1.2. The p-torsion groups T K in number theory 1.3. The logarithmic class group and Greenberg's conjecture 2. Abelian p-ramification theory for totally real fields 2.1. Main definitions and notations -The p-invariants of K 2.2. The case of the fields K = Q(ℓ n ) 2.3. General computation of the structure of T Q(ℓ n ) 2.4. Algebraic and analytic aspects 3. Definition of p-adic measures 3.1. General definition of the Stickelberger elements 3.2. Multipliers of Stickelberger elements 3.3. Spiegel involution 4. Annihilation theorem of TTest on the normalized p-adic regulator 4.5. Conjecture about the p-torsion groups T Q(ℓ n ) 5. Reflection theorem for p-class groups and p-torsion groups 5.1. Case p = 2 5.2. Case p = 2 5.3. Illustration of formula (10) of Theorem 5.3 5.4. Probabilistic analysis from the reflection theorem 6. The p-torsion groups in Q

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Section: 1mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Weber's class number problem and $p$-rationality in the cyclotomic $\widehat{\mathbb{Z}}$-extension of $\mathbb{Q}$

Gras

2020

Preprint

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“…In our previous papers [22,24,25], we proposed new methods for controlling cyclotomic units, which enabled us to prove the following theorem. …”

Section: Problem 11 Fix a Prime Number Is The Class Number H N Inmentioning

confidence: 99%

Height and Weber’s Class Number Problem

Morisawa¹,

Okazaki²

2016

J. Théor. Nombres Bordeaux

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“…If l satisfies l > (c!) 1/f , then l does not divide h 2,n for any positive integer n. THEOREM 0.3 (The case p = 3 [21]). A prime number l different from 3 is given.…”

mentioning

confidence: 99%

“…In the case p = 2, Fukuda and Komatsu [4,5,6] showed that l does not divide h 2,n for any positive integer n if l < 5 × 10 8 or l ≡ ±1 (mod 32). In the case p = 3, the first author [20,21] showed that l does not divide h 3,n for any positive integer n if l < 4 × 10 5 or l ≡ ±1 (mod 27). Moreover, in the cases p = 2 and p = 3, we improved upon Theorem 0.1: THEOREM 0.2 (The case p = 2 [22]).…”

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confidence: 99%

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

Morisawa¹,

Okazaki²

2013

Tohoku Math. J. (2)

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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?

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Mahler measure of the Horie unit and Weber's class number problem in the cyclotomic Z₃-extension of Q

Cited by 7 publications

References 1 publication

Weber's class number problem and $p$-rationality in the cyclotomic $\widehat{\mathbb{Z}}$-extension of $\mathbb{Q}$

Weber's class number problem and $p$-rationality in the cyclotomic $\widehat{\mathbb{Z}}$-extension of $\mathbb{Q}$

Height and Weber’s Class Number Problem

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

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Mahler measure of the Horie unit and Weber's class number problem in the cyclotomic Z3-extension of Q

Cited by 7 publications

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Weber's class number problem and $p$-rationality in the cyclotomic $\widehat{\mathbb{Z}}$-extension of $\mathbb{Q}$

Weber's class number problem and $p$-rationality in the cyclotomic $\widehat{\mathbb{Z}}$-extension of $\mathbb{Q}$

Height and Weber’s Class Number Problem

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

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Mahler measure of the Horie unit and Weber's class number problem in the cyclotomic Z₃-extension of Q