هندسه حسابی: محاسبات و کاربردها: شانزدهمین کنفرانس بین‌المللی در مورد هندسه حسابی، رمزنگاری و نظریه کدگذاری، 19-23 ژوئن 2017، مرکز بین‌المللی ملاقات‌های ریاضی، مارسی، فرانسه

For thirty years, the biennial international conference AGC$^2$T (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. 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 L-functions 3. Abelian varieties and surfaces 4. Generalisations 5. Theorems and conjectures of Brauer-Siegel type References 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 References Weight enumerators of Reed-Muller codes from cubic curves and their duals 1. Introduction 2. Singular projective plane cubic curves 3. Smooth projective plane cubic curves 4. Low-weight coefficients of W_{C_{2,3}^{\perp}}(X,Y) 5. Singular affine plane cubic curves 6. Smooth affine plane cubic curves 7. Low-weight coefficients of W_{(C^{A}_{2,3})^{\perp}}(X,Y) 8. Acknowledgments References The distribution of the trace in the compact group of type G2 1. Introduction 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 References The de Rham cohomology of the Suzuki curves 1. Introduction 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 References Décompositions en hauteurs locales 1. Introduction 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 1. Introduction 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 Acknowledgements References Canonical models of arithmetic (1;∞)-curves 1. Uniformizations and orders 2. \Belyi maps 3. Canonical models 4. Modular interpretations References Maps between curves and arithmetic obstructions 1. Introduction 2. The fundamental group 3. Certifying non-isomorphism 4. Examples 5. Factoring polynomials over finite fields References Back Cover