by W C Winnie Li (Pennsylvania State University)

Novel and important applications of number theory to graph theory and vice versa had been made in the past decade. The two main tools used are based on the estimates of character sums and the estimates of the eigenvalues of Hecke operators, both are rooted in the celebrated Weil conjectures settled by Deligne in 1973. The purpose of this book is to give, from scratch, a coherent and comprehensive introduction to the topics in number theory related to the central tools, with the ultimate goal of presenting their applications. This book includes many important subjects in number theory, such as Weil conjectures, Riemann-Roch theorem, L-functions, character sum estimates, modular forms, and representation theory.

• Finite Fields
• Weil Conjectures
• Local and Global Fields
• The Riemann-Roch Theorem
• Zeta and L-Functions
• Character Sum Estimates and Idèle Class Characters
• The Theory of Modular Forms
• Automorphic Forms and Automorphic Representations
• Applications