S. Lang Books

Focusing on the infinite dimensional representation theory of semisimple Lie groups, this book specifically examines SL2(R). It highlights the importance of this area in relation to fields like number theory, particularly through Langlands' work. The text aims to simplify the complexities of representation theory, which has evolved rapidly, making it challenging for newcomers. With only basic prerequisites in real analysis and differential equations, it is designed to be accessible to a broad audience, facilitating entry into this advanced subject.

Focusing on complex multiplication, this classic work by Shimura-Taniyama explores significant results in higher dimensions, building on Deuring's methods for elliptic curves. It synthesizes contributions from notable mathematicians like Weil, Serre, and Deligne, offering a clearer presentation of fundamental results compared to earlier works. The book is designed for readers familiar with abelian varieties, guiding them through key theorems and applications. It includes detailed discussions on analytic complex multiplication and the construction of abelian manifolds.

The book explores the intersection of pure arithmetic theory and algebraic-geometric theory, presenting a framework where sections function similarly to rational points. It introduces analogs of classic diophantine problems, offering a fresh perspective on algebraic systems and their properties. Through this lens, the author delves into the complexities and relationships between these mathematical concepts, providing insights into their implications and applications within the broader field of number theory.

Kummer's exploration of cyclotomic fields significantly influenced the evolution of algebraic number theory, laying the groundwork for later mathematicians like Dedekind and Hilbert. Despite its foundational role, Kummer's specific contributions were overshadowed until the mid-20th century when Iwasawa and Leopoldt revived interest in cyclotomic fields. Iwasawa's work focused on their analogues in algebraic geometry, while Leopoldt developed p-adic analogues of classical formulas, ultimately revealing a vital link between cyclotomic towers and p-adic L-functions.

Focusing on the example of SL2(R), this book explores the infinite dimensional representation theory of semisimple Lie groups. It presents the material in an accessible manner, requiring only a background in real analysis and some differential equations, making it suitable for a broad audience interested in advanced mathematical concepts.

Diophantine problems explore the existence and quantity of solutions to algebraic equations within rings and fields, focusing on integers and rational numbers. This book examines global qualitative issues, including classifications of curves with infinite integral or rational points, referencing Siegel's work and Mordell's conjecture.