Cover

Title page

Contents

Preface

Geometric error correcting codes

On products and powers of linear codes under componentwise multiplication

1. Introduction

\qquadBasic definitions

\qquadLink with tensor constructions

\qquadRank functions

\qquadGeometric aspects

2. Basic structural results and miscellaneous properties

\qquadSupport

\qquadDecomposable codes

\qquadRepeated columns

\qquadExtension of scalars

\qquadMonotonicity

\qquadStable structure

\qquadAdjunction properties

\qquadSymmetries and automorphisms

3. Estimates involving the dual distance

4. Pure bounds

\qquadThe generalized fundamental functions

\qquadAn upper bound: Singleton

\qquadLower bounds for 𝑞 large: AG codes

\qquadLower bounds for 𝑞 small: concatenation

5. Some applications

\qquadMultilinear algorithms

\qquadConstruction of lattices from codes

\qquadOblivious transfer

\qquadDecoding algorithms

\qquadAnalysis of McEliece-type cryptosystems

Appendix \thesection: A criterion for symmetric tensor decomposition

\qquadFrobenius symmetric maps

\qquadTrisymmetric and normalized multiplication algorithms

Appendix \thesection: On symmetric multilinearized polynomials

\qquadPolynomial description of symmetric powers of an extension field

\qquadEquidistributed beads on a necklace

Appendix \thesection: Review of open questions

References

Higher weights of affine Grassmann codes and their duals

1. Introduction

2. Initial higher weights

3. Terminal higher weights

4. Higher weights of duals of affine Grassmann codes

Appendix A. A geometric approach to higher weights

References

Algorithmic: special varieties

The geometry of efficient arithmetic on elliptic curves

1. Introduction

2. Background

3. Linear classification of models

4. Exact morphisms and isogenies

5. Other models for elliptic curves

6. Efficient arithmetic

References

2–2–2 isogenies between Jacobians of hyperelliptic curves

Introduction

Part 1. 2...2 isogenies and theta functions

1. Isogenies

2. Formulae

3. Classification of kernels

4. Computation of the four families

Part 2. Correspondences between family (f-2) and family (f-3)

5. Trigonal maps and trigonal construction

6. The curve C

7. A correspondence preserving hyperelliptic involutions

8. Numerical examples

References

Easy scalar decompositions for efficient scalar multiplication on elliptic curves and genus 2 Jacobians

1. Introduction

2. Relations between quadratic orders

3. General two-dimensional decompositions for elliptic curves

4. Shrinking the basis (or expanding the sublattice) to fit \G

5. Decompositions for GLV endomorphisms

6. Decompositions for the GLS endomorphism

7. Decompositions for reductions of \QQ-curves

8. Four-dimensional decompositions for GLV+GLS

9. Decompositions for the Guillevic–Ionica construction

10. Two-dimensional decompositions in genus 2

References

Algorithmic: point counting

A point counting algorithm for cyclic covers of the projective line

1. Introduction

2. Cyclic covers of the projective line

3. Monsky–Washnitzer cohomology for cyclic covers and the action of Frobenius

4. Adaptation of the Gaudry–Gürel algorithm to general cyclic covers

5. Complexity

6. Bounds on precision

7. The choice of the set of differentials

8. Numerical experiments

References

Point Counting on Non-Hyperelliptic Genus 3 Curves with Automorphism Group ℤ/2ℤ using Monsky-Washnitzer Cohomology

1. Introduction

2. Monsky-Washnitzer Cohomology

3. Cohomology of Non-Hyperelliptic Genus 3 Plane Curves with Automorphism Group ℤ/2ℤ

4. Lift of Frobenius

5. Quotient by Automorphism

6. The Algorithm

References

Wiman’s and Edge’s sextic attaining Serre’s bound II

1. Introduction

2. Wiman’s sextic I

3. Wiman’s sextic II

4. Wiman’s sextic of genus 4

5. Edge’s sextic

References

Algorithmic: general

Genetics of polynomials over local fields

Introduction

1. MacLane valuations

2. Okutsu equivalence of prime polynomials

3. Types over (𝐾,𝑣)

4. OM representations of square-free polynomials

5. Computation of the genetics of a polynomial: the Montes algorithm

6. Algorithmic applications of polynomial genetics

References

Explicit algebraic geometry

Explicit equations of optimal curves of genus 3 over certain finite fields with three parameters

1. Introduction

2. Explicit equations of plane optimal curves of genus three

References

Smooth Embeddings for the Suzuki and Ree Curves

1. Introduction

2. Preliminaries

3. The Smooth Embeddings for the Hermitian and Suzuki Curves

4. The Defining Equations and Automorphism group of the Ree Curve

5. Smooth Embedding for the Ree Curve

6. Relation to the Previous Work on the Embeddings of the Deligne–Lusztig Curves

7. Representation of the Ree Group

8. Further Properties of the Ree Curve

Appendix A. The Action of the Group 𝐺_{\fr}(𝑃_{∞}) on \Da’

References

Arithmetic geometry

Uniform distribution of zeroes of 𝐿-functions of modular forms

1. Introduction

2. Proof of theorem 1.1

References

A survey on class field theory for varieties

References

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