Annihilation of $\text{tor}_{Z_{p}}(\mathcal G_{K,S}^{ab})$ for real abelian extensions $K/Q$

Georges Gras

doi:10.33434/cams.441035

EN

Annihilation of $\text{tor}_{Z_{p}}(\mathcal G_{K,S}^{ab})$ for real abelian extensions $K/Q$

Abstract

Let $K$ be a real abelian extension of $\mathbb{Q}$. Let $p$ be a prime number, $S$ the set of $p$-places of $K$ and ${\mathcal G}_{K,S}$ the Galois group of the maximal $S \cup \{\infty\}$-ramified pro-$p$-extension of $K$ (i.e., unramified outside $p$ and $\infty$). We revisit the problem of annihilation of the $p$-torsion group ${\mathcal T}_K := \text{tor}_{Z_{p}}(\mathcal G_{K,S}^{ab})$ initiated by us and Oriat then systematized in our paper on the construction of $p$-adic $L$-functions in which we obtained a canonical ideal annihilator of ${\mathcal T}_K$ in full generality (1978--1981). Afterwards (1992--2014) some annihilators, using cyclotomic units, were proposed by Solomon, Belliard--Nguyen Quang Do, Nguyen Quang Do--Nicolas, All, Belliard--Martin. In this text, we improve our original papers and show that, in general, the Solomon elements are not optimal and/or partly degenerated. We obtain, whatever $K$ and $p$, an universal non-degenerated annihilator in terms of $p$-adic logarithms of cyclotomic numbers related to $L_p$-functions at $s=1$ of {primitive characters of $K$} (Theorem 9.4). Some computations are given with PARI programs; the case $p=2$ is analyzed and illustrated in degrees $2$, $3$, $4$ to test a conjecture.

Keywords

Abelian $p$-ramification; annihilation of $p$-torsion modules,$p$-adic $L$-functions,Stickelberger's elements,Cyclotomic units,Class field theory,Cyclotomic units

References

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Details

Primary Language

English

Subjects

Mathematical Sciences

Journal Section

Research Article

Authors

Georges Gras ^*
France

Publication Date

September 30, 2018

Submission Date

July 5, 2018

Acceptance Date

September 18, 2018

Published in Issue

Year 2018 Volume: 1 Number: 1

DOI

https://doi.org/10.33434/cams.441035

IZ

https://izlik.org/JA35SR87ZZ

APA

Gras, G. (2018). Annihilation of $\text{tor}_{Z_{p}}(\mathcal G_{K,S}^{ab})$ for real abelian extensions $K/Q$. Communications in Advanced Mathematical Sciences, 1(1), 5-34. https://doi.org/10.33434/cams.441035

AMA

1.Gras G. Annihilation of $\text{tor}_{Z_{p}}(\mathcal G_{K,S}^{ab})$ for real abelian extensions $K/Q$. Communications in Advanced Mathematical Sciences. 2018;1(1):5-34. doi:10.33434/cams.441035

Chicago

Gras, Georges. 2018. “Annihilation of $\text{tor}_{Z_{p}}(\mathcal G_{K,S}^{ab})$ for Real Abelian Extensions $K Q$”. Communications in Advanced Mathematical Sciences 1 (1): 5-34. https://doi.org/10.33434/cams.441035.

EndNote

Gras G (September 1, 2018) Annihilation of $\text{tor}_{Z_{p}}(\mathcal G_{K,S}^{ab})$ for real abelian extensions $K/Q$. Communications in Advanced Mathematical Sciences 1 1 5–34.

IEEE

[1]G. Gras, “Annihilation of $\text{tor}_{Z_{p}}(\mathcal G_{K,S}^{ab})$ for real abelian extensions $K/Q$”, Communications in Advanced Mathematical Sciences, vol. 1, no. 1, pp. 5–34, Sept. 2018, doi: 10.33434/cams.441035.

ISNAD

Gras, Georges. “Annihilation of $\text{tor}_{Z_{p}}(\mathcal G_{K,S}^{ab})$ for Real Abelian Extensions $K Q$”. Communications in Advanced Mathematical Sciences 1/1 (September 1, 2018): 5-34. https://doi.org/10.33434/cams.441035.

JAMA

1.Gras G. Annihilation of $\text{tor}_{Z_{p}}(\mathcal G_{K,S}^{ab})$ for real abelian extensions $K/Q$. Communications in Advanced Mathematical Sciences. 2018;1:5–34.

MLA

Gras, Georges. “Annihilation of $\text{tor}_{Z_{p}}(\mathcal G_{K,S}^{ab})$ for Real Abelian Extensions $K Q$”. Communications in Advanced Mathematical Sciences, vol. 1, no. 1, Sept. 2018, pp. 5-34, doi:10.33434/cams.441035.

Vancouver

1.Georges Gras. Annihilation of $\text{tor}_{Z_{p}}(\mathcal G_{K,S}^{ab})$ for real abelian extensions $K/Q$. Communications in Advanced Mathematical Sciences. 2018 Sep. 1;1(1):5-34. doi:10.33434/cams.441035

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Annihilation of $\text{tor}_{Z_{p}}(\mathcal G_{K,S}^{ab})$ for real abelian extensions $K/Q$

Abstract

Annihilation of $\text{tor}_{Z_{p}}(\mathcal G_{K,S}^{ab})$ for real abelian extensions $K/Q$

Abstract

Keywords

References

Details

Primary Language

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Journal Section

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Publication Date

Submission Date

Acceptance Date

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