Arithmetic geometry : computation and applications : 16th International Conference on Arithmetic, Geometry, Cryptography, and Coding Theory, June 19-23, 2017, Centre International de Rencontres Mathematiques, Marseille, France / Yves Aubry, Everett W. Howe, Christophe Ritzenthaler, editors.

Format:

Book

Author/Creator:

Aubry, Yves, 1965- editor.

Contributor:

Howe, Everett W., editor.

Ritzenthaler, Christophe, 1976- editor.

Series:

Contemporary mathematics (American Mathematical Society). 0271-4132 722

Contemporary mathematics, 722 0271-4132

Language:

English

Subjects (All):

Coding theory--Congresses.

Coding theory.

Geometry, Algebraic--Congresses.

Geometry, Algebraic.

Cryptography--Congresses.

Cryptography.

Number theory--Congresses.

Number theory.

Physical Description:

1 online resource (186 pages).

Edition:

1st ed.

Place of Publication:

United States of America : American Mathematical Society, 2019.

Language Note:

English

Summary:

For thirty years, the biennial international conference AGC^2T (Arithmetic, Geometry, Cryptography, and Coding Theory) has brought researchers to Marseille to build connections between arithmetic geometry and its applications, originally highlighting coding theory but more recently including cryptography and other areas as well. This volume contains the proceedings of the 16th international conference, held from June 19-23, 2017. The papers are original research articles covering a large range of topics, including weight enumerators for codes, function field analogs of the Brauer-Siegel theorem, the computation of cohomological invariants of curves, the trace distributions of algebraic groups, and applications of the computation of zeta functions of curves. Despite the varied topics, the papers share a common thread: the beautiful interplay between abstract theory and explicit results.

Contents:

Cover

Title page

Contents

Preface

Hasse-Witt and Cartier-Manin matrices: A warning and a request

Prologue

1. Matrices and semilinear algebra

2. Hasse-Witt and Cartier-Manin matrices

3. Cartier-Manin matrices for hyperelliptic curves

4. Hasse-Witt matrices through the ages

5. Subsequent developments

6. Conclusion

References

Works that cite Manin (1961) or Yui (1978)

Analogues of Brauer-Siegel theorem in arithmetic geometry

Introduction

1. Classical Brauer-Siegel theorems

2. Zeta and -functions

3. Abelian varieties and surfaces

4. Generalisations

5. Theorems and conjectures of Brauer-Siegel type

The Belyi degree of a curve is computable

1. Introduction

Acknowledgements

2. The Belyi degree

3. First proof of Theorem 1.2

4. Second proof of Theorem 1.2

5. The Fermat curve of degree four

Weight enumerators of Reed-Muller codes from cubic curves and their duals

2. Singular projective plane cubic curves

3. Smooth projective plane cubic curves

4. Low-weight coefficients of _{ _{2,3}^{\perp}}( , )

5. Singular affine plane cubic curves

6. Smooth affine plane cubic curves

7. Low-weight coefficients of _{( ^{ }_{2,3})^{\perp}}( , )

8. Acknowledgments

The distribution of the trace in the compact group of type ₂

2. Exponential sums

3. The group \Gtwo and its Lie algebra

4. Real forms

5. The Steinberg map of \Gtwo

6. Maximal torus and alcove of \UGtwo

7. The Steinberg map on \UGtwo

8. The Weyl integration formula revisited

9. Image of the alcove

10. Distribution of the trace

11. Moments

The de Rham cohomology of the Suzuki curves

2. The de Rham cohomology as a representation for the Suzuki group.

3. The Dieudonné module and de Rham cohomology

4. An explicit basis for the de Rham cohomology

Décompositions en hauteurs locales

2. Présentation des décompositions

3. Hauteurs globales, hauteurs locales

4. Modèles de Moret-Bailly des variétés abéliennes

5. Hauteur d'un point par la formule clef

6. Décomposition de la hauteur de Faltings d'une jacobienne hyperelliptique

7. Calculs explicites en dimension 1

\frenchrefname

Using zeta functions to factor polynomials over finite fields

2. Schoof's algorithm

3. Kayal's factoring idea

4. Pila's algorithm

5. Generalization of Kayal's factoring idea

6. A heuristic for Hypothesis Z

7. Weakening Hypothesis Z

8. Using varieties other than abelian varieties

Canonical models of arithmetic (1

∞)-curves

1. Uniformizations and orders

2. \Belyi maps

3. Canonical models

4. Modular interpretations

Maps between curves and arithmetic obstructions

2. The fundamental group

3. Certifying non-isomorphism

4. Examples

5. Factoring polynomials over finite fields

Back Cover.

Notes:

Includes bibliographical references.

Description based on print version record.

ISBN:

1-4704-5104-2