Cutting towers of number fields

Hajir, Farshid; Maire, Christian; Ramakrishna, Ravi

doi:10.1007/s40316-021-00156-8

Cited by 5 publications

(10 citation statements)

References 35 publications

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“…was the best known. It was later improved by Hajir and Maire [8] and finally by Hajir, Maire, and Ramakrishna [9]. The bound in question is obtained by constructing an infinite tower of fields over the totally imaginary field…”

Section: The Left Side Of the Critical Stripmentioning

confidence: 99%

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On the Northcott property of Dedekind zeta functions

Généreux¹,

Laĺın²

2022

Preprint

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The Northcott property for special values of Dedekind zeta functions and more general motivic L-functions was defined by Pazuki and Pengo. We investigate this property for any complex evaluation of Dedekind zeta functions. The results are more delicate and subtle than what was proven for the function field case in previous work of Li and the authors, since they include some surprising behavior in the neighborhood of the trivial zeros. The techniques include a mixture of analytic and computer assisted arguments.

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Section: The Left Side Of the Critical Stripmentioning

confidence: 99%

“…Theorem 1.4 is obtained by applying a result of Hajir, Maire, and Ramakrishna [9], which is an improvement of results of Martinet [11] giving upper bounds for the inferior limit of the root discriminant ∆ K…”

Section: Introductionmentioning

confidence: 99%

On the Northcott property of Dedekind zeta functions

Généreux¹,

Laĺın²

2022

Preprint

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“…Furthermore, [1, Theorem V.2.4 and Corollary V.2.4.2] give a characterization (with explicit governing fields) of the existence of degree p cyclic extensions of K with given ramification and decomposition. This criteria has been used by Hajir-Maire and Hajir-Maire-Ramakrishna in several of their papers for results on S-ramified pro-p-groups (see, e.g., [90,Theorem 5.3], [91,Remark 2.2. ] for the most recent publications).…”

Section: A54 Synthesis 2003-2005mentioning

confidence: 99%

“…In all the aspects of p-rationality that we have developed (theoretical and computational), some interesting applications are done today, including for instance, for the most recent ones, [43] by Hajir-Maire on the µ-invariant in Iwasawa's theory, then [90] by Hajir-Maire-Ramakrishna, showing the existence of p-rational fields having large p-rank of the class group, or [91] about the existence of a solvable number field L, P-ramified, whose p-Hilbert class field tower is infinite. See the bibliographies of these articles to expand the list of contributions.…”

Section: A8 Conclusion and Open Questionsmentioning

confidence: 99%

Practice of the Incomplete $p$-Ramification Over a Number Field -- History of Abelian $p$-Ramification

Gras¹

2019

Communications in Advanced Mathematical Sciences

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The theory of p-ramification, regarding the Galois group of the maximal pro-p-extension of a number field K, unramified outside p and ∞, is well known including numerical experiments with PARI/GP programs. The case of "incomplete p-ramification" (i.e., when the set S of ramified places is a strict subset of the set P of the p-places) is, on the contrary, mostly unknown in a theoretical point of view. We give, in a first part, a way to compute, for any S ⊆ P, the structure of the Galois group of the maximal S-ramified abelian pro-p-extension H K,S of any field K given by means of an irreducible polynomial. We publish PARI/GP programs usable without any special prerequisites. Then, in an Appendix, we recall the "story" of abelian S-ramification restricting ourselves to elementary aspects in order to precise much basic contributions and references, often disregarded, which may be used by specialists of other domains of number theory. Indeed, the torsion T K,S of Gal(H K,S /K) (even if S = P) is a fundamental obstruction in many problems. All relationships involving S-ramification, as Iwasawa's theory, Galois cohomology, p-adic L-functions, elliptic curves, algebraic geometry, would merit special developments, which is not the purpose of this text. Practice of the Incomplete p-Ramification Over a Number Field -History of Abelian p-Ramification -252/280The analogous theory for a local base field has also a long history that we shall not consider in this article.

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“…If one replaces the notion of p-ramification (in pro-p-extensions) by that of Σ-ramification (in pro-extensions), for any set of places Σ, the corresponding Tate-Shafarevich groups have some relations with the corresponding torsion groups T K,Σ , but with many open questions and interesting phenomena when no assumption is done (see, e.g., [32,33] for an up to date story about them and for numerical examples).…”

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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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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Cutting towers of number fields

Cited by 5 publications

References 35 publications

On the Northcott property of Dedekind zeta functions

On the Northcott property of Dedekind zeta functions

Practice of the Incomplete $p$-Ramification Over a Number Field -- History of Abelian $p$-Ramification

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

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