Frobenius Distributions : Lang-Trotter and Sato-Tate Conjectures.
Kohel, David.
Frobenius Distributions : Lang-Trotter and Sato-Tate Conjectures. - 1st ed. - 1 online resource (250 pages) - Contemporary Mathematics ; v.663 . - Contemporary Mathematics .
Cover -- Title page -- Contents -- Preface -- Lettre à Armand Borel -- Notes -- Points de repère chronologiques -- \frenchrefname -- Motivic Serre group, algebraic Sato-Tate group and Sato-Tate conjecture -- 1. Introduction -- 2. Hodge structures and Mumford-Tate group -- 3. Twisted Lefschetz groups -- 4. Hodge structures associated with -adic representations -- 5. Algebraic Sato-Tate conjecture -- 6. Connected components of \AST_ and \ST_ -- 7. Mumford-Tate group and Mumford-Tate conjecture -- 8. Some conditions for the algebraic Sato-Tate conjecture -- 9. Motivic Galois group and motivic Serre group -- 10. Motivic Mumford-Tate and Motivic Serre groups -- 11. The algebraic Sato-Tate group -- References -- An application of the effective Sato-Tate conjecture -- 1. Motivic -functions and motivic Galois groups -- 2. Equidistribution and motivic -functions -- 3. The case of an elliptic curve -- 4. The case of two elliptic curves -- 5. Notes on the general case -- Acknowledgements -- References -- Sato-Tate groups of some weight 3 motives -- 1. Introduction -- 2. Group-theoretic classification -- 3. Testing the generalized Sato-Tate conjecture -- 4. Modular forms and Hecke characters -- 5. Direct sum constructions -- 6. Tensor product constructions -- 7. The Dwork pencil -- 8. More modular constructions -- 9. Moment statistics -- Acknowledgments -- References -- Sato-Tate groups of ²= ⁸+ and ²= ⁷- . -- 1. Introduction -- 2. Background -- 3. Trace formulas -- 4. Guessing Sato-Tate groups -- 5. Determining Sato-Tate groups -- 6. Galois endomorphism types -- References -- Computing Hasse-Witt matrices of hyperelliptic curves in average polynomial time, II -- 1. Introduction -- 2. Recurrence relations -- 3. Accumulating remainder trees -- 4. Computing the first row -- 5. Hasse-Witt matrices of translated curves. 6. Computing the whole matrix -- 7. Performance results -- 8. Computing Sato-Tate distributions -- References -- Quickly constructing curves of genus 4 with many points -- 1. Introduction -- 2. A family of genus-4 curves covering a genus-2 curve -- 3. Change in defect -- 4. Interlude on work by Hayashida -- 5. Genus-2 curves with small defect -- 6. Genus-4 curves with small defect -- 7. Results -- References -- Variants of the Sato-Tate and Lang-Trotter Conjectures -- 1. Introduction -- 2. Variations of the Sato-Tate conjecture -- 3. The Lang-Trotter Conjecture on Average -- 4. Champion Primes -- References -- On the distribution of the trace in the unitary symplectic group and the distribution of Frobenius -- 1. Introduction -- 2. The unitary symplectic group -- 3. Weyl's integration formula -- 4. Equidistribution -- 5. Expressions of the law of the trace in genus 2 -- 6. The Viète map and its image -- 7. The symmetric alcove -- 8. Symmetric integration formula -- Appendix A. The character ring of -- References -- Lower-Order Biases in Elliptic Curve Fourier Coefficients in Families -- 1. Introduction -- 2. Tools for Calculating Biases -- 3. Proven Special Cases -- 4. Numerical Investigations -- 5. Conclusion and Future Work -- References -- Back Cover.
This volume contains the proceedings of the Winter School and Workshop on Frobenius Distributions on Curves, held from February 17-21, 2014 and February 24-28, 2014, at the Centre International de Rencontres Mathématiques, Marseille, France. This volume gives a representative sample of current research and developments in the rapidly developing areas of Frobenius distributions. This is mostly driven by two famous conjectures: the Sato-Tate conjecture, which has been recently proved for elliptic curves by L. Clozel, M. Harris and R. Taylor, and the Lang-Trotter conjecture, which is still widely open. Investigations in this area are based on a fine mix of algebraic, analytic and computational techniques, and the papers contained in this volume give a balanced picture of these approaches.
9781470430030
Frobenius algebras--Congresses.
Electronic books.
QA251.5.F76 2016
512.7/4
Frobenius Distributions : Lang-Trotter and Sato-Tate Conjectures. - 1st ed. - 1 online resource (250 pages) - Contemporary Mathematics ; v.663 . - Contemporary Mathematics .
Cover -- Title page -- Contents -- Preface -- Lettre à Armand Borel -- Notes -- Points de repère chronologiques -- \frenchrefname -- Motivic Serre group, algebraic Sato-Tate group and Sato-Tate conjecture -- 1. Introduction -- 2. Hodge structures and Mumford-Tate group -- 3. Twisted Lefschetz groups -- 4. Hodge structures associated with -adic representations -- 5. Algebraic Sato-Tate conjecture -- 6. Connected components of \AST_ and \ST_ -- 7. Mumford-Tate group and Mumford-Tate conjecture -- 8. Some conditions for the algebraic Sato-Tate conjecture -- 9. Motivic Galois group and motivic Serre group -- 10. Motivic Mumford-Tate and Motivic Serre groups -- 11. The algebraic Sato-Tate group -- References -- An application of the effective Sato-Tate conjecture -- 1. Motivic -functions and motivic Galois groups -- 2. Equidistribution and motivic -functions -- 3. The case of an elliptic curve -- 4. The case of two elliptic curves -- 5. Notes on the general case -- Acknowledgements -- References -- Sato-Tate groups of some weight 3 motives -- 1. Introduction -- 2. Group-theoretic classification -- 3. Testing the generalized Sato-Tate conjecture -- 4. Modular forms and Hecke characters -- 5. Direct sum constructions -- 6. Tensor product constructions -- 7. The Dwork pencil -- 8. More modular constructions -- 9. Moment statistics -- Acknowledgments -- References -- Sato-Tate groups of ²= ⁸+ and ²= ⁷- . -- 1. Introduction -- 2. Background -- 3. Trace formulas -- 4. Guessing Sato-Tate groups -- 5. Determining Sato-Tate groups -- 6. Galois endomorphism types -- References -- Computing Hasse-Witt matrices of hyperelliptic curves in average polynomial time, II -- 1. Introduction -- 2. Recurrence relations -- 3. Accumulating remainder trees -- 4. Computing the first row -- 5. Hasse-Witt matrices of translated curves. 6. Computing the whole matrix -- 7. Performance results -- 8. Computing Sato-Tate distributions -- References -- Quickly constructing curves of genus 4 with many points -- 1. Introduction -- 2. A family of genus-4 curves covering a genus-2 curve -- 3. Change in defect -- 4. Interlude on work by Hayashida -- 5. Genus-2 curves with small defect -- 6. Genus-4 curves with small defect -- 7. Results -- References -- Variants of the Sato-Tate and Lang-Trotter Conjectures -- 1. Introduction -- 2. Variations of the Sato-Tate conjecture -- 3. The Lang-Trotter Conjecture on Average -- 4. Champion Primes -- References -- On the distribution of the trace in the unitary symplectic group and the distribution of Frobenius -- 1. Introduction -- 2. The unitary symplectic group -- 3. Weyl's integration formula -- 4. Equidistribution -- 5. Expressions of the law of the trace in genus 2 -- 6. The Viète map and its image -- 7. The symmetric alcove -- 8. Symmetric integration formula -- Appendix A. The character ring of -- References -- Lower-Order Biases in Elliptic Curve Fourier Coefficients in Families -- 1. Introduction -- 2. Tools for Calculating Biases -- 3. Proven Special Cases -- 4. Numerical Investigations -- 5. Conclusion and Future Work -- References -- Back Cover.
This volume contains the proceedings of the Winter School and Workshop on Frobenius Distributions on Curves, held from February 17-21, 2014 and February 24-28, 2014, at the Centre International de Rencontres Mathématiques, Marseille, France. This volume gives a representative sample of current research and developments in the rapidly developing areas of Frobenius distributions. This is mostly driven by two famous conjectures: the Sato-Tate conjecture, which has been recently proved for elliptic curves by L. Clozel, M. Harris and R. Taylor, and the Lang-Trotter conjecture, which is still widely open. Investigations in this area are based on a fine mix of algebraic, analytic and computational techniques, and the papers contained in this volume give a balanced picture of these approaches.
9781470430030
Frobenius algebras--Congresses.
Electronic books.
QA251.5.F76 2016
512.7/4