Abelian Varieties and Number Theory

• Record Nr.: UNINA9910962811603321
• Autore: Jarden Moshe
• Pubbl/distr/stampa: Providence : , : American Mathematical Society, , 2021; ©2021
• ISBN: 9781470464233; 1470464233
• Edizione: [1st ed.]
• Descrizione fisica: 1 online resource (218 pages)
• Collana: Contemporary Mathematics ; ; v.767
• Classificazione: 14K0514A1014H0514H4014K1514H5211G0511G09
• Altri autori (Persone): ShaskaTony
• Disciplina: 516.3/53
• Soggetti: Abelian varieties; Number theory; Geometry, Algebraic; Algebraic geometry -- Abelian varieties and schemes -- Algebraic theory; Algebraic geometry -- Foundations -- Varieties and morphisms; Algebraic geometry -- Curves -- Algebraic functions; function fields [See also 11R58]; Algebraic geometry -- Curves -- Jacobians, Prym varieties [See also 32G20]; Algebraic geometry -- Abelian varieties and schemes -- Arithmetic ground fields [See also 11Dxx, 11Fxx, 11G10, 14Gxx]; Algebraic geometry -- Curves -- Elliptic curves [See also 11G05, 11G07, 14Kxx]; Number theory -- Arithmetic algebraic geometry (Diophantine geometry) [See also 11Dxx, 14Gxx, 14Kxx] -- Elliptic curves over global fields [See also 14H52]; Number theory -- Arithmetic algebraic geometry (Diophantine geometry) [See also 11Dxx, 14Gxx, 14Kxx] -- Drinfel'd modules; higher-dimensional motives, etc. [See also 14L05]
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di bibliografia: Includes bibliographical references.
• Nota di contenuto: Cover -- Title page -- Contents -- Preface -- Bibliography -- Abelian varieties and finitely generated Galois groups -- 1. Introduction -- 2. Probabilistic methods -- 3. Diophantine geometry -- 4. Combinatorics ; -- 5. Arithmetic -- 6. Analytic number theory -- Acknowledgment -- References -- Ranks of abelian varieties and the full Mordell-Lang conjecture in dimension one -- 1. Introduction -- Notation -- 2. Isotrivial and special curves -- 3. Rössler's theorem on inseparable points on curves -- 4. The Mordell-Lang conjecture -- 5. Proof of Theorem 1.1 -- 6. Corollaries and applications -- Acknowledgments -- References -- The group of automorphisms of the Heisenberg curve -- 1. Introduction -- 2. The triangle group approach -- 3. Restriction and lifting of automorphisms in covers -- 4. The Fermat curve -- 5. Automorphism group of the Heisenberg curve -- 6. Modular automorphisms -- 7. The curve ₃ -- Acknowledgment -- References -- Tate-Shafarevich groups of constant elliptic curves and isogeny volcanoes -- 1. Introduction -- 2. Preliminaries -- 3. Calculation of the Selmer group -- 4. Calculation of \Sha -- 5. Concluding remarks -- Acknowledgments -- References -- Endomorphisms of ordinary superelliptic Jacobians -- 1. Introduction -- 2. Definitions and statements -- 3. The plan of the proof -- 4. Endomorphism algebras of abelian varieties -- 5. Abelian schemes -- 6. Proof of Theorem 3.1 -- 7. Proof of Theorem 3.2 -- 8. Superelliptic Jacobians -- Acknowledgments -- References -- An introduction to Deuring's theory of constant reductions -- Introduction -- 1. The Gauss valuation -- 2. Regular reduction -- 3. The inertia theorem -- 4. Reduction of genus -- 5. Good reduction -- 6. Almost all reductions -- References -- Purely inseparable points on curves -- 1. Introduction -- 2. Semistable sheaves -- 3. Proof of Theorem 1.1 -- Acknowledgments.; References -- Correspondences on curves in positive characteristic -- 1. Introduction -- 2. Étale morphisms to the affine line -- 3. The main results and their proofs -- 4. Possible/impossible generalizations -- Acknowledgments -- References -- The inverse Galois problem over ℂ( ) -- 1. Introduction -- 2. Rings of holomorphic functions -- 3. Patching of Galois groups -- Acknowledgment -- References -- Explicit Hilbert's irreducibility theorem in function fields -- 1. Introduction -- 2. Set-up and preliminaries -- 3. Separable polynomials -- 4. Inseparable polynomials -- Acknowledgments -- References -- A comparison between obstructions to local-global principles over semi-global fields -- Introduction -- 1. Completions and henselizations of semi-global fields -- 2. Local points on -varieties -- 3. Local-global principles for rational points -- References -- The generic monodromy of Drinfeld modular varieties in special characteristic -- 1. Statement of the main result -- 2. Formal -modules, -divisible groups and deformations of Drinfeld modules -- 3. The result of Strauch -- 4. Proof of Theorem \autoref{Thm:Main} -- 5. An example -- 6. An application -- Acknowledgements -- References -- Symmetric tensor products of octonion algebras -- Introduction -- 1. The associative case -- 2. Octonion algebras -- 3. Algebras of symmetric tensors -- 4. Central idempotents -- 5. Central simple algebras -- References -- Representations of étale fundamental groups and specialization of algebraic cycles -- 1. Introduction -- 2. Representation-theoretic results -- 3. Specialization of representations of the étale fundamental group and injectivity of the restriction morphism on ¹ -- 4. Motivic representations -- 5. Specialization of the ℓ-adic Abel-Jacobi filtration on Chow groups -- 6. Applications, examples -- Acknowledgments -- References -- Back Cover.
• Sommario/riassunto: This book is a collection of articles on Abelian varieties and number theory dedicated to Gerhard Frey's 75th birthday. It contains original articles by experts in the area of arithmetic and algebraic geometry.The articles cover topics on Abelian varieties and finitely generated Galois groups, ranks of Abelian varieties and Mordell-Lang conjecture, Tate-; Shafarevich group and isogeny volcanoes, endomorphisms of superelliptic Jacobians, obstructions to local-global principles over semi-global fields, Drinfeld modular varieties, representations of etale fundamental groups and specialization of algebraic cycles, Deuring's theory of constant reductions, etc.The book will be a valuable resource to graduate students and experts working on Abelian varieties and related areas.