P-adic analysis :: a short course on recent work /
This introduction to recent work in p-adic analysis and number theory will make accessible to a relatively general audience the efforts of a number of mathematicians over the last five years. After reviewing the basics (the construction of p-adic numbers and the p-adic analog of the complex number f...
Gespeichert in:
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| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Cambridge [England] ; New York :
Cambridge University Press,
1980.
|
| Schriftenreihe: | London Mathematical Society lecture note series ;
46. |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Zusammenfassung: | This introduction to recent work in p-adic analysis and number theory will make accessible to a relatively general audience the efforts of a number of mathematicians over the last five years. After reviewing the basics (the construction of p-adic numbers and the p-adic analog of the complex number field, power series and Newton polygons), the author develops the properties of p-adic Dirichlet L-series using p-adic measures and integration. p-adic gamma functions are introduced, and their relationship to L-series is explored. Analogies with the corresponding complex analytic case are stressed. Then a formula for Gauss sums in terms of the p-adic gamma function is proved using the cohomology of Fermat and Artin-Schreier curves. Graduate students and research workers in number theory, algebraic geometry and parts of algebra and analysis will welcome this account of current research. |
| Beschreibung: | 1 online resource (163 pages) |
| Bibliographie: | Includes bibliographical references (pages 154-160) and index. |
| ISBN: | 9781107361072 1107361079 9780511526107 0511526105 |
| ISSN: | 0076-0552 |
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| 245 | 1 | 0 | |a P-adic analysis : |b a short course on recent work / |c Neal Koblitz. |
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| 490 | 1 | |a London Mathematical Society lecture note series ; |v 46, |x 0076-0552 | |
| 504 | |a Includes bibliographical references (pages 154-160) and index. | ||
| 588 | 0 | |a Print version record. | |
| 520 | |a This introduction to recent work in p-adic analysis and number theory will make accessible to a relatively general audience the efforts of a number of mathematicians over the last five years. After reviewing the basics (the construction of p-adic numbers and the p-adic analog of the complex number field, power series and Newton polygons), the author develops the properties of p-adic Dirichlet L-series using p-adic measures and integration. p-adic gamma functions are introduced, and their relationship to L-series is explored. Analogies with the corresponding complex analytic case are stressed. Then a formula for Gauss sums in terms of the p-adic gamma function is proved using the cohomology of Fermat and Artin-Schreier curves. Graduate students and research workers in number theory, algebraic geometry and parts of algebra and analysis will welcome this account of current research. | ||
| 505 | 0 | |a Cover; Half-title; Title; Copyright; Contents; Preface; CHAPTER I. BASICS; 1. History (very brief); 2. Basic concepts; 3. Power series; 4. Newton polygons; CHAPTER II. p-ADIC C-FUNCTIONS, L-FUNCTIONS AND r-FUNCTIONS; 1. Dirichlet L-series; 2. p-adic measures; 3. p-adic interpolation; 4. p-adic Dirichlet L-functions; 5. Leopoldt's formula for L (1,X); 6. The p-adic gamma function; 7. The p-adic log gamma function; 8. A formula for L'p(0,X); CHAPTER III. GAUSS SUMS AND THE p-ADIC GAMMA FUNCTION; 1. Gauss and Jacobi sums; 2. Fermat curves; 3. L-series for algebraic varieties; 4. Cohomology | |
| 505 | 8 | |a 5. p-adic cohomology6. p-adic formula for Gauss sums; 7. Stickleberger1s theorem; CHAPTER IV. p-ADIC REGULATORS; 1. Regulators and L-functions; 2. Leopoldt's p-adic regulator; 3. Gross's p-adic regulator; 4. Gross's conjecture in the abelian over Q case; APPENDIX; 1. A theorem of Amice-Fresnel; 2. The classical Stieltjes transform; 3. The Shnirelman integral and the p-adic Stieltjes transfonsfor; 4. p-adic spectral theorem; Bibliography; Index | |
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| author | Koblitz, Neal, 1948- |
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| contents | Cover; Half-title; Title; Copyright; Contents; Preface; CHAPTER I. BASICS; 1. History (very brief); 2. Basic concepts; 3. Power series; 4. Newton polygons; CHAPTER II. p-ADIC C-FUNCTIONS, L-FUNCTIONS AND r-FUNCTIONS; 1. Dirichlet L-series; 2. p-adic measures; 3. p-adic interpolation; 4. p-adic Dirichlet L-functions; 5. Leopoldt's formula for L (1,X); 6. The p-adic gamma function; 7. The p-adic log gamma function; 8. A formula for L'p(0,X); CHAPTER III. GAUSS SUMS AND THE p-ADIC GAMMA FUNCTION; 1. Gauss and Jacobi sums; 2. Fermat curves; 3. L-series for algebraic varieties; 4. Cohomology 5. p-adic cohomology6. p-adic formula for Gauss sums; 7. Stickleberger1s theorem; CHAPTER IV. p-ADIC REGULATORS; 1. Regulators and L-functions; 2. Leopoldt's p-adic regulator; 3. Gross's p-adic regulator; 4. Gross's conjecture in the abelian over Q case; APPENDIX; 1. A theorem of Amice-Fresnel; 2. The classical Stieltjes transform; 3. The Shnirelman integral and the p-adic Stieltjes transfonsfor; 4. p-adic spectral theorem; Bibliography; Index |
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| indexdate | 2024-11-27T13:25:17Z |
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| series2 | London Mathematical Society lecture note series ; |
| spelling | Koblitz, Neal, 1948- https://id.oclc.org/worldcat/entity/E39PBJvt66BCcPCt3PjPgqr8YP http://id.loc.gov/authorities/names/n77004276 P-adic analysis : a short course on recent work / Neal Koblitz. Cambridge [England] ; New York : Cambridge University Press, 1980. 1 online resource (163 pages) text txt rdacontent computer c rdamedia online resource cr rdacarrier London Mathematical Society lecture note series ; 46, 0076-0552 Includes bibliographical references (pages 154-160) and index. Print version record. This introduction to recent work in p-adic analysis and number theory will make accessible to a relatively general audience the efforts of a number of mathematicians over the last five years. After reviewing the basics (the construction of p-adic numbers and the p-adic analog of the complex number field, power series and Newton polygons), the author develops the properties of p-adic Dirichlet L-series using p-adic measures and integration. p-adic gamma functions are introduced, and their relationship to L-series is explored. Analogies with the corresponding complex analytic case are stressed. Then a formula for Gauss sums in terms of the p-adic gamma function is proved using the cohomology of Fermat and Artin-Schreier curves. Graduate students and research workers in number theory, algebraic geometry and parts of algebra and analysis will welcome this account of current research. Cover; Half-title; Title; Copyright; Contents; Preface; CHAPTER I. BASICS; 1. History (very brief); 2. Basic concepts; 3. Power series; 4. Newton polygons; CHAPTER II. p-ADIC C-FUNCTIONS, L-FUNCTIONS AND r-FUNCTIONS; 1. Dirichlet L-series; 2. p-adic measures; 3. p-adic interpolation; 4. p-adic Dirichlet L-functions; 5. Leopoldt's formula for L (1,X); 6. The p-adic gamma function; 7. The p-adic log gamma function; 8. A formula for L'p(0,X); CHAPTER III. GAUSS SUMS AND THE p-ADIC GAMMA FUNCTION; 1. Gauss and Jacobi sums; 2. Fermat curves; 3. L-series for algebraic varieties; 4. Cohomology 5. p-adic cohomology6. p-adic formula for Gauss sums; 7. Stickleberger1s theorem; CHAPTER IV. p-ADIC REGULATORS; 1. Regulators and L-functions; 2. Leopoldt's p-adic regulator; 3. Gross's p-adic regulator; 4. Gross's conjecture in the abelian over Q case; APPENDIX; 1. A theorem of Amice-Fresnel; 2. The classical Stieltjes transform; 3. The Shnirelman integral and the p-adic Stieltjes transfonsfor; 4. p-adic spectral theorem; Bibliography; Index p-adic analysis. http://id.loc.gov/authorities/subjects/sh85096399 p-adic numbers. http://id.loc.gov/authorities/subjects/sh85096402 Nombres p-adiques. Analyse p-adique. MATHEMATICS Number Theory. bisacsh p-adic numbers fast p-adic analysis fast p-adische Zahl gnd http://d-nb.info/gnd/4044292-5 Analyse p-adique. ram Print version: Koblitz, Neal, 1948- P-adic analysis. Cambridge [Eng.] ; New York : Cambridge University Press, 1980 0521280605 (DLC) 80040806 (OCoLC)6602842 London Mathematical Society lecture note series ; 46. http://id.loc.gov/authorities/names/n42015587 FWS01 ZDB-4-EBA FWS_PDA_EBA https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=552448 Volltext |
| spellingShingle | Koblitz, Neal, 1948- P-adic analysis : a short course on recent work / London Mathematical Society lecture note series ; Cover; Half-title; Title; Copyright; Contents; Preface; CHAPTER I. BASICS; 1. History (very brief); 2. Basic concepts; 3. Power series; 4. Newton polygons; CHAPTER II. p-ADIC C-FUNCTIONS, L-FUNCTIONS AND r-FUNCTIONS; 1. Dirichlet L-series; 2. p-adic measures; 3. p-adic interpolation; 4. p-adic Dirichlet L-functions; 5. Leopoldt's formula for L (1,X); 6. The p-adic gamma function; 7. The p-adic log gamma function; 8. A formula for L'p(0,X); CHAPTER III. GAUSS SUMS AND THE p-ADIC GAMMA FUNCTION; 1. Gauss and Jacobi sums; 2. Fermat curves; 3. L-series for algebraic varieties; 4. Cohomology 5. p-adic cohomology6. p-adic formula for Gauss sums; 7. Stickleberger1s theorem; CHAPTER IV. p-ADIC REGULATORS; 1. Regulators and L-functions; 2. Leopoldt's p-adic regulator; 3. Gross's p-adic regulator; 4. Gross's conjecture in the abelian over Q case; APPENDIX; 1. A theorem of Amice-Fresnel; 2. The classical Stieltjes transform; 3. The Shnirelman integral and the p-adic Stieltjes transfonsfor; 4. p-adic spectral theorem; Bibliography; Index p-adic analysis. http://id.loc.gov/authorities/subjects/sh85096399 p-adic numbers. http://id.loc.gov/authorities/subjects/sh85096402 Nombres p-adiques. Analyse p-adique. MATHEMATICS Number Theory. bisacsh p-adic numbers fast p-adic analysis fast p-adische Zahl gnd http://d-nb.info/gnd/4044292-5 Analyse p-adique. ram |
| subject_GND | http://id.loc.gov/authorities/subjects/sh85096399 http://id.loc.gov/authorities/subjects/sh85096402 http://d-nb.info/gnd/4044292-5 |
| title | P-adic analysis : a short course on recent work / |
| title_auth | P-adic analysis : a short course on recent work / |
| title_exact_search | P-adic analysis : a short course on recent work / |
| title_full | P-adic analysis : a short course on recent work / Neal Koblitz. |
| title_fullStr | P-adic analysis : a short course on recent work / Neal Koblitz. |
| title_full_unstemmed | P-adic analysis : a short course on recent work / Neal Koblitz. |
| title_short | P-adic analysis : |
| title_sort | p adic analysis a short course on recent work |
| title_sub | a short course on recent work / |
| topic | p-adic analysis. http://id.loc.gov/authorities/subjects/sh85096399 p-adic numbers. http://id.loc.gov/authorities/subjects/sh85096402 Nombres p-adiques. Analyse p-adique. MATHEMATICS Number Theory. bisacsh p-adic numbers fast p-adic analysis fast p-adische Zahl gnd http://d-nb.info/gnd/4044292-5 Analyse p-adique. ram |
| topic_facet | p-adic analysis. p-adic numbers. Nombres p-adiques. Analyse p-adique. MATHEMATICS Number Theory. p-adic numbers p-adic analysis p-adische Zahl |
| url | https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=552448 |
| work_keys_str_mv | AT koblitzneal padicanalysisashortcourseonrecentwork |