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Morisawa, Takayuki

doi:10.1016/j.jnt.2012.09.017

Cited by 3 publications

(4 citation statements)

References 21 publications

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“…We consider, in the cyclotomic Z-extension Q, any subfield of finite or infinite degree, and fix a prime p (see [50] for analytic results of non-divisibility in this context). 7.1.…”

Section: Genus Theory and P-class Groups In Qmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Gras¹

2021

Journal of Number Theory

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We revisit this problem, in a more conceptual form, since computations show that the p-torsion group TK of the Galois groups of the maximal abelian p-ramified pro-p-extension of K (Tate-Shafarevich group of K) is often non-trivial; this raises questions since # TK = # CK # RK where RK is the normalized p-adic regulator. We give a new method testing TK = 1 (Theorem 4.6, Table 6.2) and characterize the p-extensions F of K in Q with CF = 1 (Theorem 7.5 and Corollary 7.6). We publish easy to use programs, justifying again the eight known examples, and allowing further extensive computations. KTest on the normalized p-adic regulator 4.4. Conjecture about the p-torsion groups T Q(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 reflection theorem for N = ℓ n , p = 2 5.4. Probabilistic analysis for p > 2 6. The p-torsion groups in the cyclotomic Z-extension Q

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“…We consider, in the cyclotomic Z-extension Q, any subfield of finite or infinite degree, and fix a prime p (see [50] for analytic results of non-divisibility in this context). 7.1.…”

Section: Genus Theory and P-class Groups In Qmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Gras¹

2021

Journal of Number Theory

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“…Use of Genus theory. We consider, in the cyclotomic Z-extension Q of Q, composite of all the Z ℓ -extension Q(ℓ ∞ ), any subfield of degree finite or infinite, and fix a prime p (see [47] for analytic results of non-divisibility in this context). Such a field (finite or infinite) can be written K =: Q(L N ), L = {ℓ 1 , .…”

Section: 2mentioning

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%

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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