Convolution and Equidistribution (eBook)

Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

Nicholas M. Katz (Autor)

eBook Download: EPUB

2012
208 Seiten
Princeton University Press (Verlag)
978-1-4008-4270-4 (ISBN)

Infos

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.

Autor

Nicholas M. Katz is professor of mathematics at Princeton University. He is the author or coauthor of six previous titles in the Annals of Mathematics Studies: Arithmetic Moduli of Elliptic Curves (with Barry Mazur); Gauss Sums, Kloosterman Sums, and Monodromy Groups; Exponential Sums and Differential Equations; Rigid Local Systems; Twisted L-Functions and Monodromy; and Moments, Monodromy, and Perversity.

Produkt-Details

• Erscheint lt. Verlag: 24.1.2012
• Reihe/Serie: Annals of Mathematics Studies
• Reihe/Serie: Annals of Mathematics Studies
• Verlagsort: Princeton
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
• Themenwelt: Technik
• Schlagworte: Abelian category • abelian group • absolute value • Addition • Adjunction (field theory) • algebro-geometric • ArtinГchreier reduced polynomial • autoduality • autodual objects • automorphism • Big O notation • Cardinality • Change of base • characteristic two • coefficient • Commutator subgroup • Compact Group • Complex conjugate • conjugacy class • connectedness • Constant term • Contradiction • convolution • corollary • Dense set • derived category • Determinant • Diagram (category theory) • Dimension • dimensional objects • Dimension (vector space) • direct sum • discriminant • Duality • Duality (mathematics) • Eigenvalues and Eigenvectors • Emanuel Kowalski • equidistribution • Equidistribution theorem • Error Term • estimation • Euler characteristic • EulerАoincar formula • Exponential sums • fiber functor • finite field • finite field Mellin transform • Finite Fields • Finite morphism • Finite set • Fourier transform • Frobenius conjugacy • Frobenius conjugacy class • Frobenius tori • functor • fundamental group • Gauss sum • GEOM • geometrical irreducibility • GoursatЋolchinВibet theorem • Ground field • group scheme • group theory • Haar measure • Half-integer • hypergeometric sheaf • Identity component • Indecomposability • Integer • interger monic polynomials • Irreducibility (mathematics) • irreducible representation • isogenies • isomorphism class • jordan decomposition • Kloosterman sheaf • Kloosterman sum • Laurent polynomial • Legendre • Lie algebra • lie-irreducibility • linear algebra • Linear combination • Lisse • Maximal compact subgroup • mellin transform • middle convolution • middle extension sheaf • monic polynomial • Monodromy • monodromy groups • monoidal category • Monomial • Morphism • noetherian connected scheme • nonsplit form • nontrivial additive character • normal subgroup • Number Theory • odd characteristic • odd prime • One-dimensional space • orthogonal case • orthogonal group • orthogonality • Overview (debate) • Parity (mathematics) • perverse sheaf • perverse sheaves • Pierre Deligne • Point at infinity • polynomial • polynomials • Prime number • probability measure • pullback • Pullback (differential geometry) • pure weight • reductive group • residue field • Ron Evans • Root of unity • scientific notation • semidirect product • semisimple • semisimple object • Sheaf (mathematics) • Sheaves • signs • Simple Lie group • Special case • split form • subcategory • SUBGROUP • Subobject • Summation • supermorse • Symplectic Group • Tannakian category • Tannakian groups • Tensor • tensor product • Theorem • Theorems • Trace (linear algebra) • Trivial representation • uniform convergence • Unitarian trick • Unitary Representation • Unit circle • Variable (mathematics) • Verdier duality • Zeeev Rudnick
• ISBN-10: 1-4008-4270-0 / 1400842700
• ISBN-13: 978-1-4008-4270-4 / 9781400842704