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The study of exponential sums over finite fields, begun by Gauss nearly two centuries ago, has been completely transformed in recent years by advances in algebraic geometry, culminating in Deligne's work on the Weil Conjectures. It now appears as a very attractive mixture of algebraic geometry, representation theory, and the sheaf-theoretic incarnations of such standard constructions of classical analysis as convolution and Fourier transform. The book is simultaneously an account of some of these ideas, techniques, and results, and an account of their application to concrete equidistribution questions concerning Kloosterman sums and Gauss sums.

Group theory --- Algebraic geometry --- Number theory --- 511.33 --- Analytical and multiplicative number theory. Asymptotics. Sieves etc. --- 511.33 Analytical and multiplicative number theory. Asymptotics. Sieves etc. --- Gaussian sums --- Homology theory --- Kloosterman sums --- Monodromy groups --- Kloostermann sums --- Sums, Kloosterman --- Sums, Kloostermann --- Exponential sums --- Cohomology theory --- Contrahomology theory --- Algebraic topology --- Gauss sums --- Sums, Gaussian --- Analytical and multiplicative number theory. Asymptotics. Sieves etc --- Gaussian sums. --- Kloosterman sums. --- Homology theory. --- Monodromy groups. --- Number theory. --- Nombres, Théorie des. --- Exponential sums. --- Sommes exponentielles. --- Arithmetic --- Arithmétique --- Geometry, Algebraic. --- Géométrie algébrique --- Abelian category. --- Absolute Galois group. --- Absolute value. --- Additive group. --- Adjoint representation. --- Affine variety. --- Algebraic group. --- Automorphic form. --- Automorphism. --- Big O notation. --- Cartan subalgebra. --- Characteristic polynomial. --- Classification theorem. --- Coefficient. --- Cohomology. --- Cokernel. --- Combination. --- Commutator. --- Compactification (mathematics). --- Complex Lie group. --- Complex number. --- Conjugacy class. --- Continuous function. --- Convolution theorem. --- Convolution. --- Determinant. --- Diagonal matrix. --- Dimension (vector space). --- Direct sum. --- Dual basis. --- Eigenvalues and eigenvectors. --- Empty set. --- Endomorphism. --- Equidistribution theorem. --- Estimation. --- Exactness. --- Existential quantification. --- Exponential sum. --- Exterior algebra. --- Faithful representation. --- Finite field. --- Finite group. --- Four-dimensional space. --- Frobenius endomorphism. --- Fundamental group. --- Fundamental representation. --- Galois group. --- Gauss sum. --- Homomorphism. --- Integer. --- Irreducibility (mathematics). --- Isomorphism class. --- Kloosterman sum. --- L-function. --- Leray spectral sequence. --- Lie algebra. --- Lie theory. --- Maximal compact subgroup. --- Method of moments (statistics). --- Monodromy theorem. --- Monodromy. --- Morphism. --- Multiplicative group. --- Natural number. --- Nilpotent. --- Open problem. --- P-group. --- Pairing. --- Parameter space. --- Parameter. --- Partially ordered set. --- Perfect field. --- Point at infinity. --- Polynomial ring. --- Prime number. --- Quotient group. --- Representation ring. --- Representation theory. --- Residue field. --- Riemann hypothesis. --- Root of unity. --- Sheaf (mathematics). --- Simple Lie group. --- Skew-symmetric matrix. --- Smooth morphism. --- Special case. --- Spin representation. --- Subgroup. --- Support (mathematics). --- Symmetric matrix. --- Symplectic group. --- Symplectic vector space. --- Tensor product. --- Theorem. --- Trace (linear algebra). --- Trivial representation. --- Variable (mathematics). --- Weil conjectures. --- Weyl character formula. --- Zariski topology.