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Miller, John

doi:10.1016/j.jnt.2014.11.008

Cited by 7 publications

(6 citation statements)

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“…More precisely, all computations or experiments depend on the relative components T * K whose orders are given by 1 2 L p (1, ψ N ), for ψ N of order N of K. Indeed, we do not see why # C K should be always trivial for an"algebraic reason", even if it is known that R K may be, a priori, non-trivial whatever the order of magnitude of p. Moreover, an observation made in other contexts shows that, when # C * K • # R * K is non-trivial, the probability of # R * K = 1 is, roughly, p times that of # C * K = 1. The Cohen-Lenstra-Martinet heuristics (see [5,46,47] for large developments) give low probabilities for non-trivial p-class groups, even in the case of residue degree 1 of p in Q(µ N ).…”

Section: 3mentioning

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%

“…History of the main progress and results. The computation of the class number have been done in few cases because of limitation of the order of magnitude of the degree N and of p; for instance, the results given in [46,47, Tables 1, 2] only concern ℓ n = 2 7 , 3 4 , 5 2 , 11, 13, 17, 19, 23, 29, 31 (2 7 , 3 4 , 29, 31 under GRH). From PARI/GP [56], in a straightforward use, any computation needs the instruction bnfinit(P) (giving all the basic invariants of the field K defined via the polynomial P, whence the class group, a system of units, etc.…”

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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Section: 3mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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“…In [Co60], Cohn proved that h 2,3 = 1. Since then many special cases have been verified to be true [Bau69,vdLin82,Mil15,Mil14] but the conjecture still remains elusive in general. Miller has conjectured that even a stronger statement should be true [Mil15] (see also [Coa12]):…”

Section: Class Numbers Of Real Quadratic Fieldsmentioning

confidence: 99%

Class Numbers of Quadratic Fields

Bhand

Murty

2020

Hardy-Ramanujan Journal

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“…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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Class numbers in cyclotomic Zp-extensions

Cited by 7 publications

References 21 publications

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

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

Class Numbers of Quadratic Fields

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

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