Frobenius distributions : Lang-Trotter and Sato-Tate conjectures : Winter School on Frobenius Distributions on Curves, February 17-21, 2014, Workshop on Frobenius Distributions on Curves, February 24-28, 2014, Centre International de Rencontres Mathématiques, Marseille, France / David Kohel, Igor Shparlinski, editors.

Format:

Book

Contributor:

Kohel, David R., 1966- editor.

Shparlinski, Igor E., editor.

Series:

Contemporary mathematics (American Mathematical Society). 0271-4132 663

Contemporary mathematics, 663 0271-4132

Language:

English

Subjects (All):

Frobenius algebras--Congresses.

Frobenius algebras.

Curves, Algebraic--Congresses.

Curves, Algebraic.

Physical Description:

1 online resource (250 pages) : illustrations.

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2016]

Summary:

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.

Contents:

Cover

Title page

Contents

Preface

Lettre à Armand Borel

Notes

Points de repère chronologiques

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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

Sato-Tate groups of some weight 3 motives

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

Sato-Tate groups of ²= ⁸+ and ²= ⁷- .

2. Background

3. Trace formulas

4. Guessing Sato-Tate groups

5. Determining Sato-Tate groups

6. Galois endomorphism types

Computing Hasse-Witt matrices of hyperelliptic curves in average polynomial time, II

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

Quickly constructing curves of genus 4 with many points

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

Variants of the Sato-Tate and Lang-Trotter Conjectures

2. Variations of the Sato-Tate conjecture

3. The Lang-Trotter Conjecture on Average

4. Champion Primes

On the distribution of the trace in the unitary symplectic group and the distribution of Frobenius

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

Lower-Order Biases in Elliptic Curve Fourier Coefficients in Families

2. Tools for Calculating Biases

3. Proven Special Cases

4. Numerical Investigations

5. Conclusion and Future Work

Back Cover.

Notes:

Includes bibliographical references.

Description based on print version record.

ISBN:

1-4704-1947-5