Modular Forms and the Ramanujan Conjecture

This advanced graduate textbook explains the details behind fundamental geometric methods in modern number theory.

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The connection between Galois representations and modular forms has been a dominant theme in number theory in recent decades. It lies at the foundation of Deligne's proof of the classical Ramanujan Conjecture, Wiles' proof of Fermat's Last Theorem, fundamental discoveries by a number of mathematicians working on the Langlands program, and much more. In this advanced graduate textbook, a self-contained development is provided for the construction of Galois representations attached to classical modular forms, and a distinguishing feature is a careful and extensive development of the algebro-geometric machinery and techniques that one needs in this construction and its generalizations beyond the classical case. Related topics such as complex analytic spaces, Hodge structures, and etale cohomology are explained in detail and are presented in a level of generality that is suitable for considerations beyond the immediate applications to problems concerning classical modular forms. The main prerequisite is a familiarity with classical modular forms and the methods of modern algebraic geometry.
Graduate students with interests in algebraic geometry, number theory, and automorphic forms will find this book to be suitable for self-study or for mini-seminars, and researchers in number theory will find it to be a useful reference.