Introduction to Non-Abelian Class Field Theory, An: Automorphic Forms of Weight 1 and 2-Dimensional Galois Representations

Our book provides a comprehensive foray into the theoretical underpinnings and applications of non-abelian class field theory. Beginning with foundational concepts, the book gradually introduces the reader to automorphic forms of weight 1 and the intricacies of 2-dimensional Galois representations. These mathematical entities serve as pivotal components to understanding how non-commutative extensions of number fields can be comprehended through the lens of modern arithmetic geometry and representation theory.

We delve into the historical context and motivations behind non-abelian extensions, highlighting the developments that have steered modern research. By employing a methodical approach, we explore the connections between Galois groups and modular forms, which serve as the crux of much of modern number theory. Furthermore, the book elucidates on significant breakthroughs and conjectures such as the Langlands program and its implications for class field theory.

1. Foundations of Non-Abelian Class Field Theory: Gain a solid understanding of the theoretical concepts that differentiate non-abelian class field theory from its abelian counterpart.

2. Automorphic Forms and Representations: Learn about the significance of automorphic forms of weight 1 and how they play a critical role in understanding Galois representations.

3. Historical Developments: Discover the historical evolution of ideas that have influenced the current landscape of number theory and class field theories.

4. Applications and Conjectures: Explore recent mathematical conjectures and their potential implications for both theoretical and practical applications in mathematics.

1. "The nature of non-abelian groups often mirrors the complexity of the problems they aim to solve, but within that complexity lies an elegant structure waiting to be unraveled."

2. "Understanding the bridge between automorphic forms and Galois representations is akin to deciphering the harmony of a symphony composed by the numbers themselves."

3. "Every theorem and proof invites the curious mathematician to not only observe but to become intimately familiar with the dance of abstract algebra at play."

This book is essential reading for anyone seeking to deepen their understanding of modern number theory and its myriad applications. By offering insight into non-abelian class field theory, this work sheds light on an area that remains at the forefront of mathematical research. The subjects explored within are crucial for advancing theories that underpin encryption, coding theory, and even quantum computing, making this book a valuable resource for both academics and professionals alike.

Moreover, the symbiotic relationship between automorphic forms and Galois representations has profound implications across various fields of mathematics. Thus, our book not only enhances an individual's mathematical skill set but also broadens the horizon for future research and exploration. Whether you are an experienced mathematician or a graduate student, this book will offer new insights and inspire continued inquiry into the fascinating world of non-abelian class field theory.