**Contemporary Mathematics**

Volume: 663;
2016;
238 pp;
Softcover

MSC: Primary 11; 14;

**Print ISBN: 978-1-4704-1947-9
Product Code: CONM/663**

List Price: $108.00

AMS Member Price: $86.40

MAA Member Price: $97.20

**Electronic ISBN: 978-1-4704-3003-0
Product Code: CONM/663.E**

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AMS Member Price: $86.40

MAA Member Price: $97.20

# Frobenius Distributions: Lang-Trotter and Sato-Tate Conjectures

Share this page *Edited by *
*David Kohel; Igor Shparlinski*

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.

#### Readership

Graduate students and research mathematicians interested in computational aspects of number theory and the interplay between number theory and algebraic geometry.

# Table of Contents

## Frobenius Distributions: Lang-Trotter and Sato-Tate Conjectures

- Cover Cover11
- Title page iii4
- Contents v6
- Preface vii8
- Lettre à Armand Borel 110
- Motivic Serre group, algebraic Sato-Tate group and Sato-Tate conjecture 1120
- 1. Introduction 1120
- 2. Hodge structures and Mumford-Tate group 1322
- 3. Twisted Lefschetz groups 1625
- 4. Hodge structures associated with 𝑙-adic representations 1827
- 5. Algebraic Sato-Tate conjecture 2130
- 6. Connected components of \AST_{𝐾} and \ST_{𝐾} 2433
- 7. Mumford-Tate group and Mumford-Tate conjecture 3039
- 8. Some conditions for the algebraic Sato-Tate conjecture 3241
- 9. Motivic Galois group and motivic Serre group 3342
- 10. Motivic Mumford-Tate and Motivic Serre groups 3746
- 11. The algebraic Sato-Tate group 3948
- References 4251

- An application of the effective Sato-Tate conjecture 4554
- Sato-Tate groups of some weight 3 motives 5766
- 1. Introduction 5766
- 2. Group-theoretic classification 5968
- 3. Testing the generalized Sato-Tate conjecture 6372
- 4. Modular forms and Hecke characters 6978
- 5. Direct sum constructions 7382
- 6. Tensor product constructions 7685
- 7. The Dwork pencil 8089
- 8. More modular constructions 8493
- 9. Moment statistics 8695
- Acknowledgments 100109
- References 100109

- Sato-Tate groups of 𝑦²=𝑥⁸+𝑐 and 𝑦²=𝑥⁷-𝑐𝑥. 103112
- Computing Hasse–Witt matrices of hyperelliptic curves in average polynomial time, II 127136
- 1. Introduction 127136
- 2. Recurrence relations 129138
- 3. Accumulating remainder trees 131140
- 4. Computing the first row 134143
- 5. Hasse–Witt matrices of translated curves 136145
- 6. Computing the whole matrix 139148
- 7. Performance results 141150
- 8. Computing Sato–Tate distributions 144153
- References 146155

- Quickly constructing curves of genus 4 with many points 149158
- Variants of the Sato-Tate and Lang-Trotter Conjectures 175184
- On the distribution of the trace in the unitary symplectic group and the distribution of Frobenius 185194
- 1. Introduction 185194
- 2. The unitary symplectic group 187196
- 3. Weyl’s integration formula 188197
- 4. Equidistribution 191200
- 5. Expressions of the law of the trace in genus 2 196205
- 6. The Viète map and its image 203212
- 7. The symmetric alcove 207216
- 8. Symmetric integration formula 210219
- Appendix A. The character ring of 𝐺 216225
- References 220229

- Lower-Order Biases in Elliptic Curve Fourier Coefficients in Families 223232
- Back Cover Back Cover1250