Cyclotomic Fields II:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
New York, NY
Springer US
1980
|
| Schriftenreihe: | Graduate Texts in Mathematics
69 |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Beschreibung: | This second volume incorporates a number of results which were discovered and/or systematized since the first volume was being written. Again, I limit myself to the cyclotomic fields proper without introducing modular functions. As in the first volume, the main concern is with class number formulas, Gauss sums, and the like. We begin with the Ferrero-Washington theorems, proving Iwasawa's conjecture that the p-primary part of the ideal class group in the cyclotomic Zp-extension of a cyclotomic field grows linearly rather than exponentially. This is first done for the minus part (the minus referring, as usual, to the eigenspace for complex conjugation), and then it follows for the plus part because of results bounding the plus part in terms of the minus part. Kummer had already proved such results (e.g. if p,( h; then p,( h;). These are now formulated in ways applicable to the Iwasawa invariants, following Iwasawa himself. After that we do what amounts to " Dwork theory," to derive the Gross Koblitz formula expressing Gauss sums in terms of the p-adic gamma function. This lifts Stickel berger's theorem p-adically. Half of the proof relies on a course of Katz, who had first obtained Gauss sums as limits of certain factorials, and thought of using Washnitzer-Monsky cohomology to prove the Gross-Koblitz formula |
| Beschreibung: | 1 Online-Ressource |
| ISBN: | 9781468400861 9781468400885 |
| ISSN: | 0072-5285 |
| DOI: | 10.1007/978-1-4684-0086-1 |
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| 500 | |a This second volume incorporates a number of results which were discovered and/or systematized since the first volume was being written. Again, I limit myself to the cyclotomic fields proper without introducing modular functions. As in the first volume, the main concern is with class number formulas, Gauss sums, and the like. We begin with the Ferrero-Washington theorems, proving Iwasawa's conjecture that the p-primary part of the ideal class group in the cyclotomic Zp-extension of a cyclotomic field grows linearly rather than exponentially. This is first done for the minus part (the minus referring, as usual, to the eigenspace for complex conjugation), and then it follows for the plus part because of results bounding the plus part in terms of the minus part. Kummer had already proved such results (e.g. if p,( h; then p,( h;). These are now formulated in ways applicable to the Iwasawa invariants, following Iwasawa himself. After that we do what amounts to " Dwork theory," to derive the Gross Koblitz formula expressing Gauss sums in terms of the p-adic gamma function. This lifts Stickel berger's theorem p-adically. Half of the proof relies on a course of Katz, who had first obtained Gauss sums as limits of certain factorials, and thought of using Washnitzer-Monsky cohomology to prove the Gross-Koblitz formula | ||
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Datensatz im Suchindex
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| illustrated | Not Illustrated |
| indexdate | 2024-07-10T01:21:08Z |
| institution | BVB |
| isbn | 9781468400861 9781468400885 |
| issn | 0072-5285 |
| language | English |
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| publishDate | 1980 |
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| publisher | Springer US |
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| series | Graduate Texts in Mathematics |
| series2 | Graduate Texts in Mathematics |
| spelling | Lang, Serge Verfasser aut Cyclotomic Fields II by Serge Lang New York, NY Springer US 1980 1 Online-Ressource txt rdacontent c rdamedia cr rdacarrier Graduate Texts in Mathematics 69 0072-5285 This second volume incorporates a number of results which were discovered and/or systematized since the first volume was being written. Again, I limit myself to the cyclotomic fields proper without introducing modular functions. As in the first volume, the main concern is with class number formulas, Gauss sums, and the like. We begin with the Ferrero-Washington theorems, proving Iwasawa's conjecture that the p-primary part of the ideal class group in the cyclotomic Zp-extension of a cyclotomic field grows linearly rather than exponentially. This is first done for the minus part (the minus referring, as usual, to the eigenspace for complex conjugation), and then it follows for the plus part because of results bounding the plus part in terms of the minus part. Kummer had already proved such results (e.g. if p,( h; then p,( h;). These are now formulated in ways applicable to the Iwasawa invariants, following Iwasawa himself. After that we do what amounts to " Dwork theory," to derive the Gross Koblitz formula expressing Gauss sums in terms of the p-adic gamma function. This lifts Stickel berger's theorem p-adically. Half of the proof relies on a course of Katz, who had first obtained Gauss sums as limits of certain factorials, and thought of using Washnitzer-Monsky cohomology to prove the Gross-Koblitz formula Mathematics Number theory Number Theory Mathematik Graduate Texts in Mathematics 69 (DE-604)BV035421258 69 https://doi.org/10.1007/978-1-4684-0086-1 Verlag Volltext |
| spellingShingle | Lang, Serge Cyclotomic Fields II Graduate Texts in Mathematics Mathematics Number theory Number Theory Mathematik |
| title | Cyclotomic Fields II |
| title_auth | Cyclotomic Fields II |
| title_exact_search | Cyclotomic Fields II |
| title_full | Cyclotomic Fields II by Serge Lang |
| title_fullStr | Cyclotomic Fields II by Serge Lang |
| title_full_unstemmed | Cyclotomic Fields II by Serge Lang |
| title_short | Cyclotomic Fields II |
| title_sort | cyclotomic fields ii |
| topic | Mathematics Number theory Number Theory Mathematik |
| topic_facet | Mathematics Number theory Number Theory Mathematik |
| url | https://doi.org/10.1007/978-1-4684-0086-1 |
| volume_link | (DE-604)BV035421258 |
| work_keys_str_mv | AT langserge cyclotomicfieldsii |