Analytic Methods in Arithmetic Geometry

This volume contains the proceedings of the Arizona Winter School 2016, which was held from March 12-16, 2016, at The University of Arizona, Tucson, AZ. In the last decade or so, analytic methods have had great success in answering questions in arithmetic geometry and number theory. The School provi...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Bucur, Alina
Otros Autores: Zureick-Brown, David
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Providence : American Mathematical Society, 2019.
Colección:Contemporary Mathematics Ser.
Temas:
Arithmetical algebraic geometry.
Diophantine analysis.
Géométrie algébrique arithmétique.
Analyse diophantienne.
Arithmetical algebraic geometry
Diophantine analysis
Acceso en línea:Texto completo
Tabla de Contenidos:
  • Cover
  • Title page
  • Contents
  • Preface
  • Primes, elliptic curves and cyclic groups
  • 1. Introduction
  • 2. Primes
  • 3. Elliptic curves: generalities
  • 4. Elliptic curves over \Q: group structure
  • 5. Elliptic curves over \Q: division fields
  • 6. Elliptic curves over \Q: maximal Galois representations
  • 7. Elliptic curves over \Q: two-parameter families
  • 8. Elliptic curves over \Q: reductions modulo primes
  • 9. Cyclicity question: heuristics and upcoming challenges
  • 10. Cyclicity question: asymptotic
  • 11. Cyclicity question: lower bound
  • 12. Cyclicity question: average
  • 13. Primality of +1- _{ }
  • 14. Anomalous primes
  • 15. Global perspectives
  • 16. Final remarks
  • Acknowledgments
  • References
  • Growth and expansion in algebraic groups over finite fields
  • 1. Introduction
  • 2. Elementary tools
  • 3. Growth in a solvable group
  • 4. Intersections with varieties
  • 5. Growth and diameter in \SL₂()
  • 6. Further perspectives and open problems
  • Acknowledgments
  • References
  • Lectures on applied ℓ-adic cohomology
  • 1. Introduction
  • 2. Examples of trace functions
  • 3. Trace functions and Galois representations
  • 4. Summing trace functions over \Fq
  • 5. Quasi-orthogonality relations
  • 6. Trace functions over short intervals
  • 7. Autocorrelation of trace functions
  • the automorphism group of a sheaf
  • 8. Trace functions vs. primes
  • 9. Bilinear sums of trace functions
  • 10. Trace functions vs. modular forms
  • 11. The ternary divisor function in arithmetic progressions to large moduli
  • 12. The geometric monodromy group and Sato-Tate laws
  • 13. Multicorrelation of trace functions
  • 14. Advanced completion methods: the -van der Corput method
  • 15. Around Zhang's theorem on bounded gaps between primes
  • 16. Advanced completions methods: the + shift
  • Acknowledgements
  • References
  • Sato-Tate distributions
  • 1. An introduction to Sato-Tate distributions
  • 2. Equidistribution, L-functions, and the Sato-Tate conjecture for elliptic curves
  • 3. Sato-Tate groups
  • 4. Sato-Tate axioms and Galois endomorphism types
  • References
  • Back Cover

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