Genus theory of p-adic pseudo-measures -- Tame kernels and abelian p-ramification
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| Format: | Preprint |
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2023
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| _version_ | 1866915174225543168 |
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| author | Gras, Georges |
| author_facet | Gras, Georges |
| contents | We consider, for real abelian fields K, the Birch--Tate formula linking the tame kernel \#K\_2(Z\_K) to $ζ$\_K(-1); we compare, for quadratic and cyclic cubic fields with p=2,3, \#K\_2(\BZ\_K)[p^$\infty$] to the order of the torsion group T\_{K, p} of abelian p-ramification theory given by the residue of $ζ$\_{K, p}(s) at s=1. This is done via the ``genus theory'' of p-adic pseudo-measures, inaugurated in the 1970/80's and the fact that T\_{K, p} only depends on the p-class group and on the normalized p-adic regulator of K (Theorem A). We apply this to prove a conjecture of Deng--Li giving the structures of K\_2(Z\_K)[2^$\infty$] for an interesting family of real quadratic fields (Theorem B). Then, for p>3, we give a lower bound of the p-rank of K\_2(\BZ\_K) in cyclic p-extensions (Theorem C). Complements, PARI programs and tables are given in an Appendix. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_10112 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Genus theory of p-adic pseudo-measures -- Tame kernels and abelian p-ramification Gras, Georges Number Theory We consider, for real abelian fields K, the Birch--Tate formula linking the tame kernel \#K\_2(Z\_K) to $ζ$\_K(-1); we compare, for quadratic and cyclic cubic fields with p=2,3, \#K\_2(\BZ\_K)[p^$\infty$] to the order of the torsion group T\_{K, p} of abelian p-ramification theory given by the residue of $ζ$\_{K, p}(s) at s=1. This is done via the ``genus theory'' of p-adic pseudo-measures, inaugurated in the 1970/80's and the fact that T\_{K, p} only depends on the p-class group and on the normalized p-adic regulator of K (Theorem A). We apply this to prove a conjecture of Deng--Li giving the structures of K\_2(Z\_K)[2^$\infty$] for an interesting family of real quadratic fields (Theorem B). Then, for p>3, we give a lower bound of the p-rank of K\_2(\BZ\_K) in cyclic p-extensions (Theorem C). Complements, PARI programs and tables are given in an Appendix. |
| title | Genus theory of p-adic pseudo-measures -- Tame kernels and abelian p-ramification |
| topic | Number Theory |
| url | https://arxiv.org/abs/2310.10112 |