Elliptic curves and big Galois representations /:
"The mysterious properties of modular forms lie at the heart of modern number theory. This book develops a generalisation of the method of Euler systems to a two-variable deformation ring. The resulting theory is then used to study the arithmetic of elliptic curves, in particular the Birch and...
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Cambridge, UK ; New York :
Cambridge University Press,
2008.
|
| Schriftenreihe: | London Mathematical Society lecture note series ;
356. |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Zusammenfassung: | "The mysterious properties of modular forms lie at the heart of modern number theory. This book develops a generalisation of the method of Euler systems to a two-variable deformation ring. The resulting theory is then used to study the arithmetic of elliptic curves, in particular the Birch and Swinnerton-Dyer (BSD) formula." "Three main steps are outlined. The first is to parametrise 'big' cohomology groups using (deformations of) modular symbols. One can then establish finiteness results for big Selmer groups. Finally, at weight two, the arithmetic invariants of these Selmer groups allow the control of data from the BSD conjecture." "This is the first book on the subject, and the material is introduced from scratch; both graduate students and professional number theorists will find this an ideal introduction to the subject. Material at the very forefront of current research is included, and numerical examples encourage the reader to interpret abstract theorems in concrete cases."--Jacket |
| Beschreibung: | 1 online resource (ix, 281 pages) : illustrations |
| Bibliographie: | Includes bibliographical references (pages 275-279) and index. |
| ISBN: | 9781107363069 1107363063 9780511894046 051189404X 9780511721281 0511721285 9781107367975 1107367972 |
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| 245 | 1 | 0 | |a Elliptic curves and big Galois representations / |c Daniel Delbourgo. |
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| 490 | 1 | |a London Mathematical Society lecture note series ; |v 356 | |
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| 520 | 1 | |a "The mysterious properties of modular forms lie at the heart of modern number theory. This book develops a generalisation of the method of Euler systems to a two-variable deformation ring. The resulting theory is then used to study the arithmetic of elliptic curves, in particular the Birch and Swinnerton-Dyer (BSD) formula." "Three main steps are outlined. The first is to parametrise 'big' cohomology groups using (deformations of) modular symbols. One can then establish finiteness results for big Selmer groups. Finally, at weight two, the arithmetic invariants of these Selmer groups allow the control of data from the BSD conjecture." "This is the first book on the subject, and the material is introduced from scratch; both graduate students and professional number theorists will find this an ideal introduction to the subject. Material at the very forefront of current research is included, and numerical examples encourage the reader to interpret abstract theorems in concrete cases."--Jacket | |
| 588 | 0 | |a Print version record. | |
| 505 | 0 | |a Cover; Title; Copyright; Dedication; Contents; Introduction; List of Notations; Chapter I Background; 1.1 Elliptic curves; 1.2 Hasse-Weil L-functions; 1.3 Structure of the Mordell-Weil group; 1.4 The conjectures of Birch and Swinnerton-Dyer; 1.5 Modular forms and Hecke algebras; Chapter II p-Adic L-functions and Zeta Elements; 2.1 The p-adic Birch and Swinnerton-Dyer conjecture; 2.2 Perrin-Riou's local Iwasawa theory; 2.3 Integrality and (z, D)-modules; 2.4 Norm relations in K-theory; 2.5 Kato's p-adic zeta-elements; Chapter III Cyclotomic Deformations of Modular Symbols; 3.1 Q-continuity. | |
| 505 | 8 | |a 3.2 Cohomological subspaces of Euler systems3.3 The one-variable interpolation; 3.4 Local freeness of the image; Chapter IV A User's Guide to Hida Theory; 4.1 The universal ordinary Galois representation; 4.2 N-adic modular forms; 4.3 Multiplicity one for I-adic modular symbols; 4.4 Two-variable p-adic L-functions; Chapter V Crystalline Weight Deformations; 5.1 Cohomologies over deformation rings; 5.2 p-Ordinary deformations of Bcris and Dcris; 5.3 Constructing big dual exponentials; 5.4 Local dualities; Chapter VI Super Zeta-Elements; 6.1 The R-adic version of Kato's theorem. | |
| 505 | 8 | |a 6.2 A two-variable interpolation6.3 Applications to Iwasawa theory; 6.4 The Coleman exact sequence; 6.5 Computing the R[[D]]-torsion; Chapter VII Vertical and Half-Twisted Arithmetic; 7.1 Big Selmer groups; 7.2 The fundamental commutative diagrams; 7.3 Control theory for Selmer coranks; Chapter VIII Diamond-Euler Characteristics: the Local Case; 8.1 Analytic rank zero; 8.2 The Tamagawa factors away from p; 8.3 The Tamagawa factors above p (the vertical case); 8.4 The Tamagawa factors above p (the half-twisted case); 8.5 Evaluating the covolumes. | |
| 505 | 8 | |a 10.6 Numerical examples, open problemsAppendices; A: The Primitivity of Zeta Elements; B: Specialising the Universal Path Vector; C: The Weight-Variable Control Theorem (by Paul A. Smith); C.1 Notation and assumptions; C.2 Properties of affinoids; C.3 The cohomology of a lattice L; C.4 Local conditions; C.5 Dualities via the Ext-pairings; C.6 Controlling the Selmer groups; Bibliography; Index. | |
| 650 | 0 | |a Curves, Elliptic. |0 http://id.loc.gov/authorities/subjects/sh85034918 | |
| 650 | 0 | |a Galois theory. |0 http://id.loc.gov/authorities/subjects/sh85052872 | |
| 650 | 6 | |a Courbes elliptiques. | |
| 650 | 6 | |a Théorie de Galois. | |
| 650 | 7 | |a MATHEMATICS |x Geometry |x Algebraic. |2 bisacsh | |
| 650 | 7 | |a Curves, Elliptic |2 fast | |
| 650 | 7 | |a Galois theory |2 fast | |
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| 650 | 7 | |a Galois-Darstellung |2 gnd |0 http://d-nb.info/gnd/4221407-5 | |
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| contents | Cover; Title; Copyright; Dedication; Contents; Introduction; List of Notations; Chapter I Background; 1.1 Elliptic curves; 1.2 Hasse-Weil L-functions; 1.3 Structure of the Mordell-Weil group; 1.4 The conjectures of Birch and Swinnerton-Dyer; 1.5 Modular forms and Hecke algebras; Chapter II p-Adic L-functions and Zeta Elements; 2.1 The p-adic Birch and Swinnerton-Dyer conjecture; 2.2 Perrin-Riou's local Iwasawa theory; 2.3 Integrality and (z, D)-modules; 2.4 Norm relations in K-theory; 2.5 Kato's p-adic zeta-elements; Chapter III Cyclotomic Deformations of Modular Symbols; 3.1 Q-continuity. 3.2 Cohomological subspaces of Euler systems3.3 The one-variable interpolation; 3.4 Local freeness of the image; Chapter IV A User's Guide to Hida Theory; 4.1 The universal ordinary Galois representation; 4.2 N-adic modular forms; 4.3 Multiplicity one for I-adic modular symbols; 4.4 Two-variable p-adic L-functions; Chapter V Crystalline Weight Deformations; 5.1 Cohomologies over deformation rings; 5.2 p-Ordinary deformations of Bcris and Dcris; 5.3 Constructing big dual exponentials; 5.4 Local dualities; Chapter VI Super Zeta-Elements; 6.1 The R-adic version of Kato's theorem. 6.2 A two-variable interpolation6.3 Applications to Iwasawa theory; 6.4 The Coleman exact sequence; 6.5 Computing the R[[D]]-torsion; Chapter VII Vertical and Half-Twisted Arithmetic; 7.1 Big Selmer groups; 7.2 The fundamental commutative diagrams; 7.3 Control theory for Selmer coranks; Chapter VIII Diamond-Euler Characteristics: the Local Case; 8.1 Analytic rank zero; 8.2 The Tamagawa factors away from p; 8.3 The Tamagawa factors above p (the vertical case); 8.4 The Tamagawa factors above p (the half-twisted case); 8.5 Evaluating the covolumes. 10.6 Numerical examples, open problemsAppendices; A: The Primitivity of Zeta Elements; B: Specialising the Universal Path Vector; C: The Weight-Variable Control Theorem (by Paul A. Smith); C.1 Notation and assumptions; C.2 Properties of affinoids; C.3 The cohomology of a lattice L; C.4 Local conditions; C.5 Dualities via the Ext-pairings; C.6 Controlling the Selmer groups; Bibliography; Index. |
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| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/52 |
| dewey-search | 516.3/52 |
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| discipline | Mathematik |
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This book develops a generalisation of the method of Euler systems to a two-variable deformation ring. The resulting theory is then used to study the arithmetic of elliptic curves, in particular the Birch and Swinnerton-Dyer (BSD) formula." "Three main steps are outlined. The first is to parametrise 'big' cohomology groups using (deformations of) modular symbols. One can then establish finiteness results for big Selmer groups. Finally, at weight two, the arithmetic invariants of these Selmer groups allow the control of data from the BSD conjecture." "This is the first book on the subject, and the material is introduced from scratch; both graduate students and professional number theorists will find this an ideal introduction to the subject. 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| id | ZDB-4-EBA-ocn836848758 |
| illustrated | Illustrated |
| indexdate | 2024-11-27T13:25:16Z |
| institution | BVB |
| isbn | 9781107363069 1107363063 9780511894046 051189404X 9780511721281 0511721285 9781107367975 1107367972 |
| language | English |
| oclc_num | 836848758 |
| open_access_boolean | |
| owner | MAIN DE-863 DE-BY-FWS |
| owner_facet | MAIN DE-863 DE-BY-FWS |
| physical | 1 online resource (ix, 281 pages) : illustrations |
| psigel | ZDB-4-EBA |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Cambridge University Press, |
| record_format | marc |
| series | London Mathematical Society lecture note series ; |
| series2 | London Mathematical Society lecture note series ; |
| spelling | Delbourgo, Daniel. Elliptic curves and big Galois representations / Daniel Delbourgo. Cambridge, UK ; New York : Cambridge University Press, 2008. 1 online resource (ix, 281 pages) : illustrations text txt rdacontent computer c rdamedia online resource cr rdacarrier London Mathematical Society lecture note series ; 356 Includes bibliographical references (pages 275-279) and index. "The mysterious properties of modular forms lie at the heart of modern number theory. This book develops a generalisation of the method of Euler systems to a two-variable deformation ring. The resulting theory is then used to study the arithmetic of elliptic curves, in particular the Birch and Swinnerton-Dyer (BSD) formula." "Three main steps are outlined. The first is to parametrise 'big' cohomology groups using (deformations of) modular symbols. One can then establish finiteness results for big Selmer groups. Finally, at weight two, the arithmetic invariants of these Selmer groups allow the control of data from the BSD conjecture." "This is the first book on the subject, and the material is introduced from scratch; both graduate students and professional number theorists will find this an ideal introduction to the subject. Material at the very forefront of current research is included, and numerical examples encourage the reader to interpret abstract theorems in concrete cases."--Jacket Print version record. Cover; Title; Copyright; Dedication; Contents; Introduction; List of Notations; Chapter I Background; 1.1 Elliptic curves; 1.2 Hasse-Weil L-functions; 1.3 Structure of the Mordell-Weil group; 1.4 The conjectures of Birch and Swinnerton-Dyer; 1.5 Modular forms and Hecke algebras; Chapter II p-Adic L-functions and Zeta Elements; 2.1 The p-adic Birch and Swinnerton-Dyer conjecture; 2.2 Perrin-Riou's local Iwasawa theory; 2.3 Integrality and (z, D)-modules; 2.4 Norm relations in K-theory; 2.5 Kato's p-adic zeta-elements; Chapter III Cyclotomic Deformations of Modular Symbols; 3.1 Q-continuity. 3.2 Cohomological subspaces of Euler systems3.3 The one-variable interpolation; 3.4 Local freeness of the image; Chapter IV A User's Guide to Hida Theory; 4.1 The universal ordinary Galois representation; 4.2 N-adic modular forms; 4.3 Multiplicity one for I-adic modular symbols; 4.4 Two-variable p-adic L-functions; Chapter V Crystalline Weight Deformations; 5.1 Cohomologies over deformation rings; 5.2 p-Ordinary deformations of Bcris and Dcris; 5.3 Constructing big dual exponentials; 5.4 Local dualities; Chapter VI Super Zeta-Elements; 6.1 The R-adic version of Kato's theorem. 6.2 A two-variable interpolation6.3 Applications to Iwasawa theory; 6.4 The Coleman exact sequence; 6.5 Computing the R[[D]]-torsion; Chapter VII Vertical and Half-Twisted Arithmetic; 7.1 Big Selmer groups; 7.2 The fundamental commutative diagrams; 7.3 Control theory for Selmer coranks; Chapter VIII Diamond-Euler Characteristics: the Local Case; 8.1 Analytic rank zero; 8.2 The Tamagawa factors away from p; 8.3 The Tamagawa factors above p (the vertical case); 8.4 The Tamagawa factors above p (the half-twisted case); 8.5 Evaluating the covolumes. 10.6 Numerical examples, open problemsAppendices; A: The Primitivity of Zeta Elements; B: Specialising the Universal Path Vector; C: The Weight-Variable Control Theorem (by Paul A. Smith); C.1 Notation and assumptions; C.2 Properties of affinoids; C.3 The cohomology of a lattice L; C.4 Local conditions; C.5 Dualities via the Ext-pairings; C.6 Controlling the Selmer groups; Bibliography; Index. Curves, Elliptic. http://id.loc.gov/authorities/subjects/sh85034918 Galois theory. http://id.loc.gov/authorities/subjects/sh85052872 Courbes elliptiques. Théorie de Galois. MATHEMATICS Geometry Algebraic. bisacsh Curves, Elliptic fast Galois theory fast Elliptische Kurve gnd http://d-nb.info/gnd/4014487-2 Galois-Darstellung gnd http://d-nb.info/gnd/4221407-5 has work: Elliptic curves and big Galois representations (Text) https://id.oclc.org/worldcat/entity/E39PD3yK79wcBR36Jr6xV83vpP https://id.oclc.org/worldcat/ontology/hasWork Print version: Delbourgo, Daniel. Elliptic curves and big Galois representations. Cambridge, UK ; New York : Cambridge University Press, 2008 9780521728669 (DLC) 2008021192 (OCoLC)227275650 London Mathematical Society lecture note series ; 356. 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=552371 Volltext |
| spellingShingle | Delbourgo, Daniel Elliptic curves and big Galois representations / London Mathematical Society lecture note series ; Cover; Title; Copyright; Dedication; Contents; Introduction; List of Notations; Chapter I Background; 1.1 Elliptic curves; 1.2 Hasse-Weil L-functions; 1.3 Structure of the Mordell-Weil group; 1.4 The conjectures of Birch and Swinnerton-Dyer; 1.5 Modular forms and Hecke algebras; Chapter II p-Adic L-functions and Zeta Elements; 2.1 The p-adic Birch and Swinnerton-Dyer conjecture; 2.2 Perrin-Riou's local Iwasawa theory; 2.3 Integrality and (z, D)-modules; 2.4 Norm relations in K-theory; 2.5 Kato's p-adic zeta-elements; Chapter III Cyclotomic Deformations of Modular Symbols; 3.1 Q-continuity. 3.2 Cohomological subspaces of Euler systems3.3 The one-variable interpolation; 3.4 Local freeness of the image; Chapter IV A User's Guide to Hida Theory; 4.1 The universal ordinary Galois representation; 4.2 N-adic modular forms; 4.3 Multiplicity one for I-adic modular symbols; 4.4 Two-variable p-adic L-functions; Chapter V Crystalline Weight Deformations; 5.1 Cohomologies over deformation rings; 5.2 p-Ordinary deformations of Bcris and Dcris; 5.3 Constructing big dual exponentials; 5.4 Local dualities; Chapter VI Super Zeta-Elements; 6.1 The R-adic version of Kato's theorem. 6.2 A two-variable interpolation6.3 Applications to Iwasawa theory; 6.4 The Coleman exact sequence; 6.5 Computing the R[[D]]-torsion; Chapter VII Vertical and Half-Twisted Arithmetic; 7.1 Big Selmer groups; 7.2 The fundamental commutative diagrams; 7.3 Control theory for Selmer coranks; Chapter VIII Diamond-Euler Characteristics: the Local Case; 8.1 Analytic rank zero; 8.2 The Tamagawa factors away from p; 8.3 The Tamagawa factors above p (the vertical case); 8.4 The Tamagawa factors above p (the half-twisted case); 8.5 Evaluating the covolumes. 10.6 Numerical examples, open problemsAppendices; A: The Primitivity of Zeta Elements; B: Specialising the Universal Path Vector; C: The Weight-Variable Control Theorem (by Paul A. Smith); C.1 Notation and assumptions; C.2 Properties of affinoids; C.3 The cohomology of a lattice L; C.4 Local conditions; C.5 Dualities via the Ext-pairings; C.6 Controlling the Selmer groups; Bibliography; Index. Curves, Elliptic. http://id.loc.gov/authorities/subjects/sh85034918 Galois theory. http://id.loc.gov/authorities/subjects/sh85052872 Courbes elliptiques. Théorie de Galois. MATHEMATICS Geometry Algebraic. bisacsh Curves, Elliptic fast Galois theory fast Elliptische Kurve gnd http://d-nb.info/gnd/4014487-2 Galois-Darstellung gnd http://d-nb.info/gnd/4221407-5 |
| subject_GND | http://id.loc.gov/authorities/subjects/sh85034918 http://id.loc.gov/authorities/subjects/sh85052872 http://d-nb.info/gnd/4014487-2 http://d-nb.info/gnd/4221407-5 |
| title | Elliptic curves and big Galois representations / |
| title_auth | Elliptic curves and big Galois representations / |
| title_exact_search | Elliptic curves and big Galois representations / |
| title_full | Elliptic curves and big Galois representations / Daniel Delbourgo. |
| title_fullStr | Elliptic curves and big Galois representations / Daniel Delbourgo. |
| title_full_unstemmed | Elliptic curves and big Galois representations / Daniel Delbourgo. |
| title_short | Elliptic curves and big Galois representations / |
| title_sort | elliptic curves and big galois representations |
| topic | Curves, Elliptic. http://id.loc.gov/authorities/subjects/sh85034918 Galois theory. http://id.loc.gov/authorities/subjects/sh85052872 Courbes elliptiques. Théorie de Galois. MATHEMATICS Geometry Algebraic. bisacsh Curves, Elliptic fast Galois theory fast Elliptische Kurve gnd http://d-nb.info/gnd/4014487-2 Galois-Darstellung gnd http://d-nb.info/gnd/4221407-5 |
| topic_facet | Curves, Elliptic. Galois theory. Courbes elliptiques. Théorie de Galois. MATHEMATICS Geometry Algebraic. Curves, Elliptic Galois theory Elliptische Kurve Galois-Darstellung |
| url | https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=552371 |
| work_keys_str_mv | AT delbourgodaniel ellipticcurvesandbiggaloisrepresentations |