Complex multiplication:
This is a self-contained 2010 account of the state of the art in classical complex multiplication that includes recent results on rings of integers and applications to cryptography using elliptic curves. The author is exhaustive in his treatment, giving a thorough development of the theory of ellipt...
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Cambridge
Cambridge University Press
2010
|
| Schriftenreihe: | New mathematical monographs
15 |
| Schlagworte: | |
| Online-Zugang: | BSB01 FHN01 Volltext |
| Zusammenfassung: | This is a self-contained 2010 account of the state of the art in classical complex multiplication that includes recent results on rings of integers and applications to cryptography using elliptic curves. The author is exhaustive in his treatment, giving a thorough development of the theory of elliptic functions, modular functions and quadratic number fields and providing a concise summary of the results from class field theory. The main results are accompanied by numerical examples, equipping any reader with all the tools and formulas they need. Topics covered include: the construction of class fields over quadratic imaginary number fields by singular values of the modular invariant j and Weber's tau-function; explicit construction of rings of integers in ray class fields and Galois module structure; the construction of cryptographically relevant elliptic curves over finite fields; proof of Berwick's congruences using division values of the Weierstrass p-function; relations between elliptic units and class numbers |
| Beschreibung: | Title from publisher's bibliographic system (viewed on 05 Oct 2015) |
| Beschreibung: | 1 online resource (xiii, 361 pages) |
| ISBN: | 9780511776892 |
| DOI: | 10.1017/CBO9780511776892 |
Internformat
MARC
| LEADER | 00000nmm a2200000zcb4500 | ||
|---|---|---|---|
| 001 | BV043941691 | ||
| 003 | DE-604 | ||
| 005 | 00000000000000.0 | ||
| 007 | cr|uuu---uuuuu | ||
| 008 | 161206s2010 |||| o||u| ||||||eng d | ||
| 020 | |a 9780511776892 |c Online |9 978-0-511-77689-2 | ||
| 024 | 7 | |a 10.1017/CBO9780511776892 |2 doi | |
| 035 | |a (ZDB-20-CBO)CR9780511776892 | ||
| 035 | |a (OCoLC)839035485 | ||
| 035 | |a (DE-599)BVBBV043941691 | ||
| 040 | |a DE-604 |b ger |e rda | ||
| 041 | 0 | |a eng | |
| 049 | |a DE-12 |a DE-92 | ||
| 082 | 0 | |a 516.3/52 |2 22 | |
| 084 | |a SK 180 |0 (DE-625)143222: |2 rvk | ||
| 084 | |a SK 240 |0 (DE-625)143226: |2 rvk | ||
| 100 | 1 | |a Schertz, Reinhard |d 1943- |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Complex multiplication |c Reinhard Schertz |
| 264 | 1 | |a Cambridge |b Cambridge University Press |c 2010 | |
| 300 | |a 1 online resource (xiii, 361 pages) | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b c |2 rdamedia | ||
| 338 | |b cr |2 rdacarrier | ||
| 490 | 0 | |a New mathematical monographs |v 15 | |
| 500 | |a Title from publisher's bibliographic system (viewed on 05 Oct 2015) | ||
| 505 | 8 | |a Preface -- 1. Elliptic functions -- 2. Modular functions -- 3. Basic facts from number theory -- 4. Factorisation of singular values -- 5. The reciprocity law -- 6. Generation of ring class fields and ray class fields -- 7. Integral basis in ray class fields -- 8. Galois module structure -- 9. Berwick's congruences -- 10. Cryptographically relevant elliptic curves -- 11. The class number formulas of Curt Meyer -- 12. Arithmetic interpretation of class number formulas -- References -- Index of notation -- Index | |
| 520 | |a This is a self-contained 2010 account of the state of the art in classical complex multiplication that includes recent results on rings of integers and applications to cryptography using elliptic curves. The author is exhaustive in his treatment, giving a thorough development of the theory of elliptic functions, modular functions and quadratic number fields and providing a concise summary of the results from class field theory. The main results are accompanied by numerical examples, equipping any reader with all the tools and formulas they need. Topics covered include: the construction of class fields over quadratic imaginary number fields by singular values of the modular invariant j and Weber's tau-function; explicit construction of rings of integers in ray class fields and Galois module structure; the construction of cryptographically relevant elliptic curves over finite fields; proof of Berwick's congruences using division values of the Weierstrass p-function; relations between elliptic units and class numbers | ||
| 650 | 4 | |a Multiplication, Complex | |
| 650 | 0 | 7 | |a Modulfunktion |0 (DE-588)4039855-9 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Komplexe Multiplikation |0 (DE-588)4164903-5 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Elliptische Funktion |0 (DE-588)4134665-8 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Komplexe Multiplikation |0 (DE-588)4164903-5 |D s |
| 689 | 0 | 1 | |a Elliptische Funktion |0 (DE-588)4134665-8 |D s |
| 689 | 0 | 2 | |a Modulfunktion |0 (DE-588)4039855-9 |D s |
| 689 | 0 | |8 1\p |5 DE-604 | |
| 776 | 0 | 8 | |i Erscheint auch als |n Druckausgabe |z 978-0-521-76668-5 |
| 776 | 0 | 8 | |i Erscheint auch als |n Druckausgabe |z 978-1-107-47177-1 |
| 856 | 4 | 0 | |u https://doi.org/10.1017/CBO9780511776892 |x Verlag |z URL des Erstveröffentlichers |3 Volltext |
| 912 | |a ZDB-20-CBO | ||
| 999 | |a oai:aleph.bib-bvb.de:BVB01-029350661 | ||
| 883 | 1 | |8 1\p |a cgwrk |d 20201028 |q DE-101 |u https://d-nb.info/provenance/plan#cgwrk | |
| 966 | e | |u https://doi.org/10.1017/CBO9780511776892 |l BSB01 |p ZDB-20-CBO |q BSB_PDA_CBO |x Verlag |3 Volltext | |
| 966 | e | |u https://doi.org/10.1017/CBO9780511776892 |l FHN01 |p ZDB-20-CBO |q FHN_PDA_CBO |x Verlag |3 Volltext | |
Datensatz im Suchindex
| _version_ | 1804176883754991616 |
|---|---|
| any_adam_object | |
| author | Schertz, Reinhard 1943- |
| author_facet | Schertz, Reinhard 1943- |
| author_role | aut |
| author_sort | Schertz, Reinhard 1943- |
| author_variant | r s rs |
| building | Verbundindex |
| bvnumber | BV043941691 |
| classification_rvk | SK 180 SK 240 |
| collection | ZDB-20-CBO |
| contents | Preface -- 1. Elliptic functions -- 2. Modular functions -- 3. Basic facts from number theory -- 4. Factorisation of singular values -- 5. The reciprocity law -- 6. Generation of ring class fields and ray class fields -- 7. Integral basis in ray class fields -- 8. Galois module structure -- 9. Berwick's congruences -- 10. Cryptographically relevant elliptic curves -- 11. The class number formulas of Curt Meyer -- 12. Arithmetic interpretation of class number formulas -- References -- Index of notation -- Index |
| ctrlnum | (ZDB-20-CBO)CR9780511776892 (OCoLC)839035485 (DE-599)BVBBV043941691 |
| dewey-full | 516.3/52 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/52 |
| dewey-search | 516.3/52 |
| dewey-sort | 3516.3 252 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| doi_str_mv | 10.1017/CBO9780511776892 |
| format | Electronic eBook |
| fullrecord | <?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>03697nmm a2200541zcb4500</leader><controlfield tag="001">BV043941691</controlfield><controlfield tag="003">DE-604</controlfield><controlfield tag="005">00000000000000.0</controlfield><controlfield tag="007">cr|uuu---uuuuu</controlfield><controlfield tag="008">161206s2010 |||| o||u| ||||||eng d</controlfield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">9780511776892</subfield><subfield code="c">Online</subfield><subfield code="9">978-0-511-77689-2</subfield></datafield><datafield tag="024" ind1="7" ind2=" "><subfield code="a">10.1017/CBO9780511776892</subfield><subfield code="2">doi</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(ZDB-20-CBO)CR9780511776892</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(OCoLC)839035485</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-599)BVBBV043941691</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-604</subfield><subfield code="b">ger</subfield><subfield code="e">rda</subfield></datafield><datafield tag="041" ind1="0" ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-12</subfield><subfield code="a">DE-92</subfield></datafield><datafield tag="082" ind1="0" ind2=" "><subfield code="a">516.3/52</subfield><subfield code="2">22</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">SK 180</subfield><subfield code="0">(DE-625)143222:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">SK 240</subfield><subfield code="0">(DE-625)143226:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Schertz, Reinhard</subfield><subfield code="d">1943-</subfield><subfield code="e">Verfasser</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Complex multiplication</subfield><subfield code="c">Reinhard Schertz</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="a">Cambridge</subfield><subfield code="b">Cambridge University Press</subfield><subfield code="c">2010</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">1 online resource (xiii, 361 pages)</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="b">c</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="b">cr</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="490" ind1="0" ind2=" "><subfield code="a">New mathematical monographs</subfield><subfield code="v">15</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">Title from publisher's bibliographic system (viewed on 05 Oct 2015)</subfield></datafield><datafield tag="505" ind1="8" ind2=" "><subfield code="a">Preface -- 1. Elliptic functions -- 2. Modular functions -- 3. Basic facts from number theory -- 4. Factorisation of singular values -- 5. The reciprocity law -- 6. Generation of ring class fields and ray class fields -- 7. Integral basis in ray class fields -- 8. Galois module structure -- 9. Berwick's congruences -- 10. Cryptographically relevant elliptic curves -- 11. The class number formulas of Curt Meyer -- 12. Arithmetic interpretation of class number formulas -- References -- Index of notation -- Index</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">This is a self-contained 2010 account of the state of the art in classical complex multiplication that includes recent results on rings of integers and applications to cryptography using elliptic curves. The author is exhaustive in his treatment, giving a thorough development of the theory of elliptic functions, modular functions and quadratic number fields and providing a concise summary of the results from class field theory. The main results are accompanied by numerical examples, equipping any reader with all the tools and formulas they need. Topics covered include: the construction of class fields over quadratic imaginary number fields by singular values of the modular invariant j and Weber's tau-function; explicit construction of rings of integers in ray class fields and Galois module structure; the construction of cryptographically relevant elliptic curves over finite fields; proof of Berwick's congruences using division values of the Weierstrass p-function; relations between elliptic units and class numbers</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Multiplication, Complex</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Modulfunktion</subfield><subfield code="0">(DE-588)4039855-9</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Komplexe Multiplikation</subfield><subfield code="0">(DE-588)4164903-5</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Elliptische Funktion</subfield><subfield code="0">(DE-588)4134665-8</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="689" ind1="0" ind2="0"><subfield code="a">Komplexe Multiplikation</subfield><subfield code="0">(DE-588)4164903-5</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2="1"><subfield code="a">Elliptische Funktion</subfield><subfield code="0">(DE-588)4134665-8</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2="2"><subfield code="a">Modulfunktion</subfield><subfield code="0">(DE-588)4039855-9</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2=" "><subfield code="8">1\p</subfield><subfield code="5">DE-604</subfield></datafield><datafield tag="776" ind1="0" ind2="8"><subfield code="i">Erscheint auch als</subfield><subfield code="n">Druckausgabe</subfield><subfield code="z">978-0-521-76668-5</subfield></datafield><datafield tag="776" ind1="0" ind2="8"><subfield code="i">Erscheint auch als</subfield><subfield code="n">Druckausgabe</subfield><subfield code="z">978-1-107-47177-1</subfield></datafield><datafield tag="856" ind1="4" ind2="0"><subfield code="u">https://doi.org/10.1017/CBO9780511776892</subfield><subfield code="x">Verlag</subfield><subfield code="z">URL des Erstveröffentlichers</subfield><subfield code="3">Volltext</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">ZDB-20-CBO</subfield></datafield><datafield tag="999" ind1=" " ind2=" "><subfield code="a">oai:aleph.bib-bvb.de:BVB01-029350661</subfield></datafield><datafield tag="883" ind1="1" ind2=" "><subfield code="8">1\p</subfield><subfield code="a">cgwrk</subfield><subfield code="d">20201028</subfield><subfield code="q">DE-101</subfield><subfield code="u">https://d-nb.info/provenance/plan#cgwrk</subfield></datafield><datafield tag="966" ind1="e" ind2=" "><subfield code="u">https://doi.org/10.1017/CBO9780511776892</subfield><subfield code="l">BSB01</subfield><subfield code="p">ZDB-20-CBO</subfield><subfield code="q">BSB_PDA_CBO</subfield><subfield code="x">Verlag</subfield><subfield code="3">Volltext</subfield></datafield><datafield tag="966" ind1="e" ind2=" "><subfield code="u">https://doi.org/10.1017/CBO9780511776892</subfield><subfield code="l">FHN01</subfield><subfield code="p">ZDB-20-CBO</subfield><subfield code="q">FHN_PDA_CBO</subfield><subfield code="x">Verlag</subfield><subfield code="3">Volltext</subfield></datafield></record></collection> |
| id | DE-604.BV043941691 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T07:39:16Z |
| institution | BVB |
| isbn | 9780511776892 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-029350661 |
| oclc_num | 839035485 |
| open_access_boolean | |
| owner | DE-12 DE-92 |
| owner_facet | DE-12 DE-92 |
| physical | 1 online resource (xiii, 361 pages) |
| psigel | ZDB-20-CBO ZDB-20-CBO BSB_PDA_CBO ZDB-20-CBO FHN_PDA_CBO |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | Cambridge University Press |
| record_format | marc |
| series2 | New mathematical monographs |
| spelling | Schertz, Reinhard 1943- Verfasser aut Complex multiplication Reinhard Schertz Cambridge Cambridge University Press 2010 1 online resource (xiii, 361 pages) txt rdacontent c rdamedia cr rdacarrier New mathematical monographs 15 Title from publisher's bibliographic system (viewed on 05 Oct 2015) Preface -- 1. Elliptic functions -- 2. Modular functions -- 3. Basic facts from number theory -- 4. Factorisation of singular values -- 5. The reciprocity law -- 6. Generation of ring class fields and ray class fields -- 7. Integral basis in ray class fields -- 8. Galois module structure -- 9. Berwick's congruences -- 10. Cryptographically relevant elliptic curves -- 11. The class number formulas of Curt Meyer -- 12. Arithmetic interpretation of class number formulas -- References -- Index of notation -- Index This is a self-contained 2010 account of the state of the art in classical complex multiplication that includes recent results on rings of integers and applications to cryptography using elliptic curves. The author is exhaustive in his treatment, giving a thorough development of the theory of elliptic functions, modular functions and quadratic number fields and providing a concise summary of the results from class field theory. The main results are accompanied by numerical examples, equipping any reader with all the tools and formulas they need. Topics covered include: the construction of class fields over quadratic imaginary number fields by singular values of the modular invariant j and Weber's tau-function; explicit construction of rings of integers in ray class fields and Galois module structure; the construction of cryptographically relevant elliptic curves over finite fields; proof of Berwick's congruences using division values of the Weierstrass p-function; relations between elliptic units and class numbers Multiplication, Complex Modulfunktion (DE-588)4039855-9 gnd rswk-swf Komplexe Multiplikation (DE-588)4164903-5 gnd rswk-swf Elliptische Funktion (DE-588)4134665-8 gnd rswk-swf Komplexe Multiplikation (DE-588)4164903-5 s Elliptische Funktion (DE-588)4134665-8 s Modulfunktion (DE-588)4039855-9 s 1\p DE-604 Erscheint auch als Druckausgabe 978-0-521-76668-5 Erscheint auch als Druckausgabe 978-1-107-47177-1 https://doi.org/10.1017/CBO9780511776892 Verlag URL des Erstveröffentlichers Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Schertz, Reinhard 1943- Complex multiplication Preface -- 1. Elliptic functions -- 2. Modular functions -- 3. Basic facts from number theory -- 4. Factorisation of singular values -- 5. The reciprocity law -- 6. Generation of ring class fields and ray class fields -- 7. Integral basis in ray class fields -- 8. Galois module structure -- 9. Berwick's congruences -- 10. Cryptographically relevant elliptic curves -- 11. The class number formulas of Curt Meyer -- 12. Arithmetic interpretation of class number formulas -- References -- Index of notation -- Index Multiplication, Complex Modulfunktion (DE-588)4039855-9 gnd Komplexe Multiplikation (DE-588)4164903-5 gnd Elliptische Funktion (DE-588)4134665-8 gnd |
| subject_GND | (DE-588)4039855-9 (DE-588)4164903-5 (DE-588)4134665-8 |
| title | Complex multiplication |
| title_auth | Complex multiplication |
| title_exact_search | Complex multiplication |
| title_full | Complex multiplication Reinhard Schertz |
| title_fullStr | Complex multiplication Reinhard Schertz |
| title_full_unstemmed | Complex multiplication Reinhard Schertz |
| title_short | Complex multiplication |
| title_sort | complex multiplication |
| topic | Multiplication, Complex Modulfunktion (DE-588)4039855-9 gnd Komplexe Multiplikation (DE-588)4164903-5 gnd Elliptische Funktion (DE-588)4134665-8 gnd |
| topic_facet | Multiplication, Complex Modulfunktion Komplexe Multiplikation Elliptische Funktion |
| url | https://doi.org/10.1017/CBO9780511776892 |
| work_keys_str_mv | AT schertzreinhard complexmultiplication |