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

Georges GRAS

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

Article Details

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

Authors:

Georges GRAS

Publisher:

Emrah Evren KARA

Main Topics:

Mathematics

Journal:

Communications in Advanced Mathematical Sciences 2018 Vol.1 Issue 1, pp.5-34

Document Type:

Article

Publication Date:

9/2018

Document Language:

English

Document ID (Org No.):

2000911752

Frequency:

Quarterly

pISSN:

2651-4001

DOI:

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

Abstract:

 


Let KKK be a real abelian extension of QQ\mathbb{Q}. Let ppp be a prime number, SSS the set of ppp-places of KKK and GK,SGK,S{\mathcal G}_{K,S} the Galois group of the maximal S∪{∞}S∪{∞}S \cup \{\infty\}-ramified pro-ppp-extension of KKK (i.e., unramified outside ppp and ∞∞\infty). We revisit the problem of annihilation of the ppp-torsion group TK:=torZp(GabK,S)TK:=torZp(GK,Sab){\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 ppp-adic LLL-functions in which we obtained a canonical ideal annihilator of TKTK{\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 KKK and ppp, an universal non-degenerated annihilator in terms of ppp-adic logarithms of cyclotomic numbers related to LpLpL_p-functions at s=1s=1s=1 of {primitive characters of KKK} (Theorem 9.4). Some computations are given with PARI programs; the case p=2p=2p=2 is analyzed and illustrated in degrees 222, 333, 444 to test a conjecture.

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Article Details

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

Authors: Georges GRAS

Publisher: Emrah Evren KARA

Main Topics: Mathematics

Journal: Communications in Advanced Mathematical Sciences 2018 Vol.1 Issue 1, pp.5-34

Document Type: Article

Publication Date: 9/2018

Document Language: English

Document ID (Org No.): 2000911752

Frequency: Quarterly

pISSN: 2651-4001

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

Issues

Authors:

Georges GRAS

Publisher:

Emrah Evren KARA

Main Topics:

Mathematics

Journal:

Communications in Advanced Mathematical Sciences 2018 Vol.1 Issue 1, pp.5-34

Document Type:

Article

Publication Date:

9/2018

Document Language:

English

Document ID (Org No.):

2000911752

Frequency:

Quarterly

pISSN:

2651-4001

DOI:

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