حسابداری، هندسه، رمزنگاری و نظریه کدگذاری: کنفرانس سیزدهم حسابداری، هندسه، رمزنگاری و نظریه کدگذاری Cirm، مارسی، فرانسه، ... فرانسه، 19 ژوئن

This volume contains the proceedings of the 13th $\mathrm{AGC^2T}$ conference, held March 14–18, 2011, in Marseille, France, together with the proceedings of the 2011 Geocrypt conference, held June 19–24, 2011, in Bastia, France. The original research articles contained in this volume cover various topics ranging from algebraic number theory to Diophantine geometry, curves and abelian varieties over finite fields and applications to codes, boolean functions or cryptography. The international conference $\mathrm{AGC^2T}$, which is held every two years in Marseille, France, has been a major event in the area of applied arithmetic geometry for more than 25 years. Preface 8 Construction of a \Bbbk-complete addition law on Jacobians of hyperelliptic curves of genus two 10 1. Introduction 10 2. Construction 16 Appendix A. Operation count 20 References 21 Number of points in an Artin-Schreier covering 24 Introduction 24 1. Number of points in Artin-Schreier coverings: Weil bound 27 2. Study of an etale sheaf attached to a family of exponential sums 28 3. Cohomology spaces, and the proof of the theorem 32 4. Global monodromy, and a refinement of the theorem 33 References 35 Some more functions that are not APN infinitely often. The case of Gold and Kasami exponents 36 1. Introduction 36 2. Preliminaries 37 3. Some functions that are not APN infinitely often 38 4. Polynomials of Kasami degree 39 5. Binomials that are not APN infinitely often 42 References 44 Rational curves with many rational points over a finite field 46 1. Introduction 46 2. Arithmetic of the curve B 47 3. Geometry of B with F_{q}-lines, for q odd 51 4. Geometry of B with F_{q}-lines, for q even 52 5. Codes from Ballico-Hefez curves 53 6. Generalization of the curve B 54 References 57 Enumeration of Splitting Subspaces over Finite Fields 58 1. Introduction 58 2. Easy cases and guesses 59 3. Splitting planes 60 4. Refinements and Extensions 62 Appendix A. Vector Recurrences and Singer Cycles 64 References 66 The characteristic polynomials of abelian varieties of dimension 4 over finite fields 68 1. Introduction and results 68 2. The coefficients of Weil polynomials of degree 8 70 3. Newton polygons 73 4. Supersingular case 76 Acknowledgements 76 References 77 New bounds on the maximum number of points on genus-4 curves over small finite fields 78 1. Introduction 78 2. Restrictions on genus-4 curves with many points 80 3. Double covers of elliptic curves 80 4. Hermitian forms over maximal quadratic orders 83 5. Hermitian forms over nonmaximal quadratic orders 85 6. Hermitian forms over Z[ζ5] 89 7. Lower bounds from explicit examples 93 References 94 Some planar maps and related function fields 96 1. Introduction 96 2. Proving that Fu(x) is planar for some u∈F_{q3}* 98 3. Related function fields and proving that Fu(x) is not planar for some u∈F_{q3}* 103 4. Completing the classification of planar maps Fu(x) 110 Acknowledgments 122 References 122 New families of APN functions in characteristic 3 or 5 124 1. Introduction 124 2. Proof of conjectures 1.4 and 1.5 126 3. Proof of theorem 1.9 130 4. Some remarks about Zha and Wang theorems 131 References 132 Identities for Kloosterman sums and modular curves 134 1. Background 134 2. Kloosterman sum identities 135 Acknowledgment 138 References 138 Degree growth, linear independence and periods of a class of rational dynamical systems 140 1. Introduction 140 2. Structure of the Iterations 143 3. Degree Growth 144 4. Linear Independence 145 5. Trajectory Lengths 146 6. Maximal Periods 147 References 151 Computer search for curves with many points among abelian covers of genus 2 curves 154 Background 154 Description of the method 154 Implementation and results 156 References 158 The groups of points on abelian surfaces over finite fields 160 1. Introduction 160 2. Main result 161 3. Matrix factorizations 163 4. Proof of case 3 165 References 166 Computing low-degree isogenies in genus 2 with the Dolgachev–Lehavi method 168 1. Introduction 168 2. An overview of the Dolgachev–Lehavi construction 169 3. The domain curve 169 4. The kernel of the isogeny 169 5. The rational normal curve 170 6. The secant lines 171 7. The Weierstrass subspace 172 8. The theorem of Dolgachev and Lehavi 173 9. From theory to practice 174 10. The codomain curve 175 11. The algorithm for l=3 176 12. The algorithm in practice 178 References 179 Hodge classes on certain hyperelliptic prymians 180 1. Definitions and statements 180 2. Galois theory 182 3. Homomorphisms of hyperelliptic jacobians 184 4. Hodge groups of hyperelliptic jacobians 187 5. Prym varieties 189 References 191