Half-integral weight modular forms and real quadratic $p$-rational fields

Assim, Jilali; Bouazzaoui, Zakariae

doi:10.7169/facm/1851

Cited by 10 publications

(11 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As we get at most x 2α terms which are O x 1/2 (log x) 2 (under GRH), this error term is O x 1 2 +2α (log x) 2 . Thus, we take α such that ε < α < 1 4 − ε (justifying the fact that we took ε < 1 8 ), so that with A(x) → +∞, which is sufficient to prove the proposition.…”

Section: A a Stronger Analytic Proposition Under Grhmentioning

confidence: 99%

Triquadratic p-Rational Fields

Koperecz¹

2021

Preprint

View full text Add to dashboard Cite

In his work about Galois representations, Greenberg conjectured the existence, for any odd prime p and any positive integer t, of a multiquadratic p-rational number field of degree 2 t . In this article, we prove that there exists infinitely many primes p such that the triquadratic fieldTo do this, we use an analytic result, proved apart in section §4, providing us with infinitely many prime numbers p such that p + 2 et p − 2 have "big" square factors. Therefore the related imaginary quadratic subfieldshave "small" discriminants for infinitely many primes p. In the spirit of Brauer-Siegel estimates, it proves that the class numbers of these imaginary quadratic fields are relatively prime to p, and so prove their p-rationality.

show abstract

Section: A a Stronger Analytic Proposition Under Grhmentioning

confidence: 99%

Triquadratic p-Rational Fields

Koperecz¹

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Remark 3.5. A proof of the existence of infinitely many real quadratic 5-rational fields using half-integral weight modular forms can be found in [1].…”

Section: < Pmentioning

confidence: 99%

“…where o k denotes the ring of integers of k. Accordingly ε ∈ U (1) v \ U (2) v where again U (i) v denotes the i-th higher unit group in the local field k v for a p-adic prime v of k. This guarantees the fact that our unit ε (hence, also the principal unit) is not a p-th power locally at v. Let us now look at the class number h k . It is possible to prove h k < p using again [23, Corollary 2] but here we have a better upper bound.…”

Section: Real Bi-quadratic Casementioning

confidence: 99%

Multi-quadratic $p$-rational Number Fields

Benmerieme¹,

Movahhedi²

2020

Preprint

View full text Add to dashboard Cite

For each odd prime p, we prove the existence of infinitely many real quadratic fields which are p-rational. Explicit imaginary and real bi-quadratic p-rational fields are also given for each prime p. Using a recent method developed by Greenberg, we deduce the existence of Galois extensions of Q with Galois group isomorphic to an open subgroup of GLn(Zp), for n = 4 and n = 5 and at least for all the primes p < 192.699.943.

show abstract

“…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

View full text Add to dashboard Cite

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.

show abstract

Half-integral weight modular forms and real quadratic $p$-rational fields

Cited by 10 publications

References 12 publications

Triquadratic p-Rational Fields

Triquadratic p-Rational Fields

Multi-quadratic $p$-rational Number Fields

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

Contact Info

Product

Resources

About