Indivisibility of special values of Dedekind zeta functions of real quadratic fields

Byeon, Dongho

doi:10.4064/aa109-3-3

Cited by 7 publications

(10 citation statements)

References 3 publications

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“…In general, given an odd prime number p, it is not known wether there exist infinitely many real quadratic fields which are p-rational. This is known for the cases of p = 3 which is proved by Dongho Byeon in [B,Theorem 1], and the other case is p = 5 (see [A-B]). Both cases are proved using divisibility properties of Fourier coefficients of half-integer weight modular forms.…”

Section: Williams Congruencementioning

confidence: 81%

Fibonacci numbers and real quadratic p-rational fields

Bouazzaoui

2020

Period Math Hung

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Section: Williams Congruencementioning

confidence: 81%

Fibonacci numbers and real quadratic p-rational fields

Bouazzaoui

2020

Period Math Hung

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“…The following program gives, as t grows from 1 up to B, the Kummer radical M and the integer r obtained from the factorizations of m 1 (t) = t 2 − 1, under the form Mr 2 ; then we put them in a list LM and the F.O.P. algorithm gives the pairs C = core(mt, 1) = [M, r], in the increasing order of the radicals M and removes the duplicate entries: [2,2], [3,1], [5,4], [6,2], [7,3], [10,6], [11,3], [13,180], [14,4], [15,1], [17,8], [19,39], [21,12], [22,42], [23,5], [26,10], [29,1820], [30,2], [31,273], [33,4], [34,6], [35,1], [37,12], [38,6], [39,4], ...…”

Section: The Imaginary Cyclic Extensionmentioning

confidence: 99%

Unlimited Lists of Quadratic Integers of Given Norm Application to Some Arithmetic Properties

GRAS

2023

Communications in Advanced Mathematical Sciences

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We use the polynomials $m_s(t) = t^2 - 4 s$, $s \in \{-1, 1\}$, in an elementary process giving unlimited lists of {\it fundamental units of norm $s$}, of real quadratic fields, with ascending order of the discriminants. As $t$ grows from $1$ to an upper bound $\BB$, for each {\it first occurrence} of a square-free integer $M \geq 2$, in the factorization $m_s(t) =: M r^2$, the unit $\frac{1}{2} \big(t + r \sqrt{M}\big)$ is the fundamental unit of norm $s$ of $\Q(\sqrt M)$, even if $r >1$ (Theorem \ref{fondunit}). Using $m_{s\nu}(t) = t^2 - 4 s \nu$, $\nu \geq 2$, the algorithm gives unlimited lists of {\it fundamental integers of norm $s\nu$} (Theorem~\ref{unicity}). We deduce, for any prime $p>2$, unlimited lists of {\it non $p$-rational} quadratic fields (Theorems \ref{prational}, \ref{heuristic}, \ref{infty}) and lists of degree $p-1$ imaginary fields with non-trivial $p$-class group (Theorems \ref{cubic}, \ref{quintic}). All PARI programs are given.

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“…In another viewpoint, as in [115] (after some works of Hartung, Horie, Naito) and [116], it is shown, using modular forms, the infiniteness of p-rational real quadratic fields for p = 3 and p = 5. These are the only solutions for p < 10 8 .…”

Section: A64 Order Of Magnitude Of T Kp and Conjecturesmentioning

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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Indivisibility of special values of Dedekind zeta functions of real quadratic fields

Cited by 7 publications

References 3 publications

Fibonacci numbers and real quadratic p-rational fields

Fibonacci numbers and real quadratic p-rational fields

Unlimited Lists of Quadratic Integers of Given Norm Application to Some Arithmetic Properties

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

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