Arithmetic, Geometry, Cryptography, and Coding Theory 2021: 18th International Conference Arithmetic, Geometry, Cryptography, and Coding Theory May 31 ... France (Contemporary Mathematics, 779)

✍ Scribed by Samuele Anni (editor), Valentijn Karemaker (editor), Elisa Lorenzo Garcia (editor)

• Publisher: AMS
• Year: 2022
• Tongue: English
• Leaves: 198
• Category: Library

✦ Synopsis

The May-June 2021 conference was planned for Marseilles, France, but was held online because of the COVID-19 pandemic. The 10 papers in the proceedings examine arithmetic, geometry, cryptography, and coding theory from such perspectives as a strategy to optimize the complexity of Chudnobsky-type algorithms over the projective line, the constant D(q) defined by Homma, arithmetic monodromy groups of dynamical Belyi maps, automorphisms and isogeny graphs of abelian varieties with applications to the superspecial Richelot isogeny graph, and the regulator dominates the rank. Annotation ©2022 Ringgold, Inc., Portland, OR (protoview.com)

✦ Table of Contents

Cover
Title page
Contents
Preface
Numerical reconstruction of curves from their Jacobians
1. Introduction
2. The Dubrovin threefold
3. Numerical recovery
Acknowledgments
References
A strategy to optimize the complexity of Chudnovsky-type algorithms over the projective line
1. Introduction
2. Chudnovsky-type algorithms
3. Optimization of scalar complexity
4. Examples
Acknowledgment
References
On the constant 𝐷(𝑞) defined by Homma
1. Introduction
2. An upper bound for 𝐷(𝑞): the proof of Item 1 in Theorem 1.5
3. A lower bound for 𝐷(𝑞): the proof of Item 2 in Theorem 1.5
4. A lower bound for 𝐷(𝑞²): the proof of Item 3 in Theorem 1.5
References
How big is the image of the Galois representations attached to CM elliptic curves?
1. Introduction
2. Analogues of Serre’s open image theorem for CM elliptic curves
3. A formula for the index
4. How to compute the index in practice
5. Explicit examples
Acknowledgments
References
Multiradical isogenies
1. Introduction
2. Background
3. On the existence of multiradical isogeny formulae
4. Examples
5. Multiradical (3,3)-isogenies
6. Hash function from (3,3)-isogenies
Appendix: code for 3-torsion
Acknowledgments
References
Arithmetic monodromy groups of dynamical Belyi maps
1. Introduction
2. Automorphism group of 𝑇
3. Belyi Maps
4. Monodromy groups of dynamical Belyi maps
5. Normalizer of 𝐸 and 𝑈 inside 𝑊
6. Arithmetic monodromy groups of dynamical Belyi maps
References
Automorphisms and isogeny graphs of abelian varieties, with applications to the superspecial Richelot isogeny graph
1. Introduction
2. Isogeny graphs
3. Isogenies and automorphisms
4. Random walks
5. The Richelot isogeny graph
6. Random walks in the superspecial Richelot isogeny graph
7. Connectivity and diameters
8. An example: the superspecial Richelot graph for 𝑝=47
Appendix A. Experimental diameters and 𝜆_{⋆} for Γ^{𝑆𝑆}₂(2;𝑝)
Appendix B. Explicit formulæ for genus-2 computations
References
Frobenius structures on hypergeometric equations
1. Introduction
2. Generalities
3. Hypergeometric equations and the GKZ construction
4. Hypergeometric Frobenius intertwiners
5. Applications to computation of 𝐿-functions
6. Towards 𝐴-hypergeometric motives
References
The regulator dominates the rank
1. Introduction
2. Definitions and prerequisites
3. Regulators of elliptic curves over function fields of positive characteristic
Acknowledgments
References
Introduction to Drinfeld modules
1. Applications
2. Analytic theory
3. Algebraic theory
4. Reduction theory
5. Example: The Carlitz module
6. Class field theory
7. Drinfeld modular varieties
Acknowledgments
References
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