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Convolution and Equidistribution explores an important aspect of number theory--the theory of exponential sums over finite fields and their Mellin transforms--from a new, categorical point of view. The book presents fundamentally important results and a plethora of examples, opening up new directions in the subject. The finite-field Mellin transform (of a function on the multiplicative group of a finite field) is defined by summing that function against variable multiplicative characters. The basic question considered in the book is how the values of the Mellin transform are distributed (in a probabilistic sense), in cases where the input function is suitably algebro-geometric. This question is answered by the book's main theorem, using a mixture of geometric, categorical, and group-theoretic methods. By providing a new framework for studying Mellin transforms over finite fields, this book opens up a new way for researchers to further explore the subject.

Mellin transform. --- Convolutions (Mathematics) --- Sequences (Mathematics) --- Mathematical sequences --- Numerical sequences --- Algebra --- Mathematics --- Convolution transforms --- Transformations, Convolution --- Distribution (Probability theory) --- Functions --- Integrals --- Transformations (Mathematics) --- Transform, Mellin --- Integral transforms --- ArtinГchreier reduced polynomial. --- Emanuel Kowalski. --- EulerАoincar formula. --- Frobenius conjugacy class. --- Frobenius conjugacy. --- Frobenius tori. --- GoursatЋolchinВibet theorem. --- Kloosterman sheaf. --- Laurent polynomial. --- Legendre. --- Pierre Deligne. --- Ron Evans. --- Tannakian category. --- Tannakian groups. --- Zeeev Rudnick. --- algebro-geometric. --- autodual objects. --- autoduality. --- characteristic two. --- connectedness. --- dimensional objects. --- duality. --- equidistribution. --- exponential sums. --- fiber functor. --- finite field Mellin transform. --- finite field. --- finite fields. --- geometrical irreducibility. --- group scheme. --- hypergeometric sheaf. --- interger monic polynomials. --- isogenies. --- lie-irreducibility. --- lisse. --- middle convolution. --- middle extension sheaf. --- monic polynomial. --- monodromy groups. --- noetherian connected scheme. --- nonsplit form. --- nontrivial additive character. --- number theory. --- odd characteristic. --- odd prime. --- orthogonal case. --- perverse sheaves. --- polynomials. --- pure weight. --- semisimple object. --- semisimple. --- sheaves. --- signs. --- split form. --- supermorse. --- theorem. --- theorems.