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Transcendence and Linear Relations of 1-Periods [Kietas viršelis]

(Eidgenössische Technische Hochschule Zürich), (Albert-Ludwigs-Universität Freiburg, Germany)
  • Formatas: Hardback, 263 pages, aukštis x plotis x storis: 235x157x20 mm, weight: 500 g, Worked examples or Exercises
  • Serija: Cambridge Tracts in Mathematics
  • Išleidimo metai: 26-May-2022
  • Leidėjas: Cambridge University Press
  • ISBN-10: 1316519937
  • ISBN-13: 9781316519936
Kitos knygos pagal šią temą:
Transcendence and Linear Relations of 1-PeriodsDidesnis paveikslėlis
  • Formatas: Hardback, 263 pages, aukštis x plotis x storis: 235x157x20 mm, weight: 500 g, Worked examples or Exercises
  • Serija: Cambridge Tracts in Mathematics
  • Išleidimo metai: 26-May-2022
  • Leidėjas: Cambridge University Press
  • ISBN-10: 1316519937
  • ISBN-13: 9781316519936
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9781316519936.html
Raktažodžiai:
This exploration of the relation between periods and transcendental numbers brings Baker's theory of linear forms in logarithms into its most general framework, the theory of 1-motives. Written by leading experts in the field, it contains original results and finalises the theory of linear relations of 1-periods, answering long-standing questions in transcendence theory. It provides a complete exposition of the new theory for researchers, but also serves as an introduction to transcendence for graduate students and newcomers. It begins with foundational material, including a review of the theory of commutative algebraic groups and the analytic subgroup theorem as well as the basics of singular homology and de Rham cohomology. Part II addresses periods of 1-motives, linking back to classical examples like the transcendence of p, before the authors turn to periods of algebraic varieties in Part III. Finally, Part IV aims at a dimension formula for the space of periods of a 1-motive in terms of its data.

This work answers long-standing open questions in transcendence theory and finalises the theory of linear relations of 1-periods. It serves as a detailed and modern introduction for graduate students and young researchers to the beautiful world of transcendence. The authors include foundational material and link examples back to classical results.

Recenzijos

' the book under review is surely a foundational work, which finally settles many open conjectures involving periods of curves. It has also the merit of providing references and proofs for a vast amount of foundational material, including many variants of the theory of motives. As such, it will surely become a standard reference for many works to come.' Riccardo Pengo, zbMATH Open

Daugiau informacijos

Leading experts explore the relation between periods and transcendental numbers, using a modern approach derived from the theory of motives.
Prologue xi
Acknowledgements xx
1 Introduction
1(12)
1.1 Transcendence
1(1)
1.2 Relations Between Periods
2(2)
1.3 Dimensions of Period Spaces
4(3)
1.4 Method of Proof
7(1)
1.5 Why I-Motives?
8(1)
1.6 The Case of Elliptic Curves
8(1)
1.7 Values of Hypergeometric Functions
9(1)
1.8 Structure of the Monograph
10(3)
PART ONE FOUNDATIONS
13(54)
2 Basics on Categories
15(6)
2.1 Additive and Ahelian Categories
15(2)
2.2 Subcategories
17(1)
2.3 Functors
18(3)
3 Homology and Cohomology
21(10)
3.1 Singular Homology
21(3)
3.2 Algebraic de Rham Cohomology
24(5)
3.3 The Period Pairing
29(2)
4 Commutative Algebraic Groups
31(12)
4.1 The Building Blocks
31(2)
4.2 Group Extensions
33(4)
4.3 Semi-abelian Varieties
37(2)
4.4 Universal Vector Extensions
39(1)
4.5 Generalised Jacobians
40(3)
5 Lie Groups
43(5)
5.1 The Lie Algebra
43(1)
5.2 The Exponential Map
44(1)
5.3 Integration over Paths
45(3)
6 The Analytic Subgroup Theorem
48(6)
6.1 The Statement
48(2)
6.2 Analytic vs Algebraic Homomorphisms
50(4)
7 The Formalism of the Period Conjecture
54(13)
7.1 Periods
54(5)
7.2 The Period Conjecture
59(8)
PART TWO PERIODS OF DELIGNE 1-MOTIVES
67(42)
8 Deligne's 1-Motives
69(13)
8.1 The Category and the Realisation Functors
69(6)
8.2 The Functor to Mixed Hodge Structures
75(4)
8.3 The Key Comparison
79(3)
9 Periods of 1-Motives
82(11)
9.1 Definition and First Properties
82(2)
9.2 Relations Between Periods
84(3)
9.3 Transcendence of Periods of 1-Motives
87(2)
9.4 Fullness
89(4)
10 First Examples
93(8)
10.1 Squaring the Circle
93(2)
10.2 Transcendence of Logarithms
95(2)
10.3 Hilbert's Seventh Problem
97(2)
10.4 Abelian Periods for Closed Paths
99(2)
11 On Non-closed Elliptic Periods
101(8)
11.1 The Setting
101(1)
11.2 Without CM
102(2)
11.3 The CM-Case
104(2)
11.4 Transcendence
106(3)
PART THREE PERIODS OF ALGEBRAIC VARIETIES
109(36)
12 Periods of Algebraic Varieties
111(9)
12.1 Spaces of Cohomological 1-Periods
111(1)
12.2 Periods of Curve Type
112(3)
12.3 Comparison with Periods of 1-Motives
115(2)
12.4 The Motivic Point of View
117(3)
13 Relations Between Periods
120(8)
13.1 Kontsevich's Period Conjecture
120(3)
13.2 The Case of Curves
123(5)
14 Vanishing of Periods of Curves
128(17)
14.1 Classical Periods
128(2)
14.2 The Setting
130(4)
14.3 Forms of the First Kind
134(2)
14.4 Forms of the Second Kind
136(2)
14.5 Forms of the Third Kind
138(2)
14.6 Arbitrary Differential Forms
140(1)
14.7 Vanishing of Simple Periods
141(4)
PART FOUR DIMENSIONS OF PERIOD SPACES
145(62)
15 Dimension Computations: An Estimate
147(16)
15.1 Set-up and Terminology
147(3)
15.2 The Saturated Case
150(5)
15.3 Special Cases
155(4)
15.4 Proof of the Dimension Estimate
159(4)
16 Structure of the Period Space
163(4)
17 Incomplete Periods of the Third Kind
167(11)
17.1 Relation Spaces
167(7)
17.2 Alternative Description of σinc3(M)
174(4)
18 Elliptic Curves
178(13)
18.1 Classical Theory of Periods
178(3)
18.2 Elliptic Periods
181(2)
18.3 A Calculation
183(1)
18.4 Transcendence of Incomplete Periods
184(2)
18.5 Elliptic Period Space
186(5)
19 Values of Hypergeometric Functions
191(16)
19.1 Elliptic Integrals
191(6)
19.2 Abelian Integrals
197(10)
PART FIVE APPENDICES
207(22)
Appendix A Nori Motives
209(8)
Appendix B Voevodsky Motives
217(4)
Appendix C Comparison of Realisations
221(8)
List of Notation 229(6)
References 235(6)
Index 241
Annette Huber is Professor for Number Theory at Albert-Ludwigs-Universität Freiburg. She works in arithmetic geometry and is a leading specialist in the theory of motives. Together with Stefan Müller-Stach, she authored the book Periods and Nori motives (2017). She was a speaker at the 2002 ICM and is a member of the German National Academy of Sciences, the Leopoldina. Gisbert Wüstholz is Professor Emeritus at ETH Zurich. He is a leading researcher in transcendence theory and diophantine geometry. In 1986, he was an invited speaker at the ICM in Berkeley, in 1992 he gave the Mordell Lecture and in 2001 the Kuwait Foundation Lecture. He is Honorary Professor at Togji University Shanghai and at TU Graz. He is an elected member of four academies including the Leopoldina and has published six books.