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Modular forms are tremendously important in various areas of mathematics, from number theory and algebraic geometry to combinatorics and lattices. Their Fourier coefficients, with Ramanujan's tau-function as a typical example, have deep arithmetic significance. Prior to this book, the fastest known algorithms for computing these Fourier coefficients took exponential time, except in some special cases. This book gives an algorithm for computing coefficients of modular forms of level one in polynomial time. For example, Ramanujan's tau of a prime number p can be computed in time bounded by a fixed power of the logarithm of p. Such fast computation of Fourier coefficients is itself based on the main result of the book: the computation, in polynomial time, of Galois representations over finite fields attached to modular forms by the Langlands program. Because these Galois representations typically have a nonsolvable image, this result is a major step forward from explicit class field theory, and it could be described as the start of the explicit Langlands program. The book begins with a concise and concrete introduction that makes it accessible to readers without an extensive background in arithmetic geometry, and it includes a chapter that describes actual computations.

This book presents an important breakthrough in arithmetic geometry. In 2014, this book's author delivered a series of lectures at the University of California, Berkeley, on new ideas in the theory of p-adic geometry. Building on his discovery of perfectoid spaces, the author introduced the concept of “diamonds,” which are to perfectoid spaces what algebraic spaces are to schemes. The introduction of diamonds, along with the development of a mixed-characteristic shtuka, set the stage for a critical advance in the discipline. This book shows that the moduli space of mixed-characteristic shtukas is a diamond, raising the possibility of using the cohomology of such spaces to attack the Langlands conjectures for a reductive group over a p-adic field. The book follows the informal style of the original Berkeley lectures, with one chapter per lecture. It explores p-adic and perfectoid spaces before laying out the newer theory of shtukas and their moduli spaces. Points of contact with other threads of the subject, including p-divisible groups, p-adic Hodge theory, and Rapoport-Zink spaces, are thoroughly explained.