Arithmetic Geometry, Number Theory, and Computation

This volume contains articles related to the Simons Collaboration's work in arithmetic geometry, number theory, and computation, which aim to develop systematic tools for analyzing Diophantine properties of curves, surfaces, and abelian varieties over number fields and finite fields.

Format: Paperback / softback
Length: 587 pages
Publication date: 16 March 2023
Publisher: Springer Nature Switzerland AG

This extensive volume encompasses a collection of articles that delve into the realm of research conducted by the esteemed Simons Collaboration, focusing on the fields of "Arithmetic Geometry, Number Theory, and Computation." The papers presented within this publication offer invaluable mathematical insights and innovative algorithms that are crucial for the development of large-scale databases such as the L-functions and Modular Forms Database (LMFDB). The authors of these articles aspire to establish comprehensive methodologies for analyzing the Diophantine properties of curves, surfaces, and abelian varieties across number fields and finite fields. Furthermore, the articles explore examples that hold significant potential for future research endeavors. Some of the specific topics covered in this volume include:
1. Algebraic varieties over finite fields: This section explores the study of algebraic varieties, which are mathematical structures defined over finite fields. It discusses various techniques and methods for studying these varieties, including the Chabauty-Coleman method, modular forms, and rational points on curves of small genus.
2. The Chabauty-Coleman method: This method is a powerful tool in the study of modular forms, which are special functions that arise in number theory and have applications in various fields, such as cryptography and computer science. The Chabauty-Coleman method involves the computation of certain L-functions, which are related to modular forms, and has played a significant role in the development of arithmetic geometry.
3. Modular forms: Modular forms are mathematical functions that exhibit periodic behavior and are important in number theory and cryptography. This section discusses the theory of modular forms, their properties, and their applications in various fields, including the study of elliptic curves and the construction of cryptographic protocols.
4. Rational points on curves of small genus: This section explores the study of rational points on curves of small genus, which are points on curves that satisfy certain algebraic equations. It discusses the methods and techniques used to find rational points, as well as their applications in number theory and cryptography.
5. S-unit equations and integral points: This section discusses the study of S-unit equations, which are equations involving the sum of two squares of rational functions. It also explores the theory of integral points, which are points on curves that satisfy certain integral equations. These topics have important applications in number theory and cryptography.
Overall, this volume provides a comprehensive overview of the latest developments in the fields of arithmetic geometry, number theory, and computation, and is an essential resource for researchers and scholars in these areas.

Weight: 908g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030809164
Edition number: 1st ed. 2021