Download Convolution and Equidistribution: Sato-Tate Theorems for by Nicholas M. Katz PDF

By Nicholas M. Katz

Convolution and Equidistribution explores a big element of quantity theory--the thought of exponential sums over finite fields and their Mellin transforms--from a brand new, express viewpoint. The publication provides essentially very important effects and a plethora of examples, commencing up new instructions within the topic. The finite-field Mellin rework (of a functionality at the multiplicative workforce of a finite box) is outlined through summing that functionality opposed to variable multiplicative characters. the fundamental query thought of within the booklet is how the values of the Mellin remodel are disbursed (in a probabilistic sense), in circumstances the place the enter functionality is definitely algebro-geometric. this question is spoke back by means of the book's major theorem, utilizing a mix of geometric, specific, and group-theoretic tools. by way of delivering a brand new framework for learning Mellin transforms over finite fields, this ebook opens up a brand new manner for researchers to additional discover the topic.

Show description

Read or Download Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms PDF

Best algebraic geometry books

Lectures on Algebraic Geometry 1: Sheaves, Cohomology of Sheaves, and Applications to Riemann Surfaces (Aspects of Mathematics, Volume 35)

This publication and the subsequent moment quantity is an advent into glossy algebraic geometry. within the first quantity the tools of homological algebra, thought of sheaves, and sheaf cohomology are constructed. those equipment are fundamental for contemporary algebraic geometry, yet also they are primary for different branches of arithmetic and of serious curiosity of their personal.

Spaces of Homotopy Self-Equivalences: A Survey

This survey covers teams of homotopy self-equivalence periods of topological areas, and the homotopy kind of areas of homotopy self-equivalences. For manifolds, the entire team of equivalences and the mapping category crew are in comparison, as are the corresponding areas. incorporated are equipment of calculation, various calculations, finite new release effects, Whitehead torsion and different components.

Coding Theory and Algebraic Geometry: Proceedings of the International Workshop held in Luminy, France, June 17-21, 1991

Approximately ten years in the past, V. D. Goppa came upon a shocking connection among the concept of algebraic curves over a finite box and error-correcting codes. the purpose of the assembly "Algebraic Geometry and Coding concept" used to be to provide a survey at the current kingdom of study during this box and comparable subject matters.

Algorithms in algebraic geometry

Within the final decade, there was a burgeoning of task within the layout and implementation of algorithms for algebraic geometric compuation. a few of these algorithms have been initially designed for summary algebraic geometry, yet now are of curiosity to be used in functions and a few of those algorithms have been initially designed for purposes, yet now are of curiosity to be used in summary algebraic geometry.

Additional resources for Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms

Example text

We argue by contradiction. If N is not punctual, it has some arithmetically irreducible constituent M which is not punctual. Then Ggeom,M is finite, being a quotient of Ggeom,N . So we are reduced to the case when M is arithmetically irreducible, of the form G[1] for an arithmetically irreducible middle extension sheaf G. We wish to reduce further to the case in which G is geometrically irreducible. Think of G as the extension by direct image of an arithmetically irreducible lisse sheaf F on a dense open set U ⊂ G.

N ). Then χ(A1 /k, j0 ! N ) = χc (A1 /k, j0 ! N ) because χ = χc on a curve (and indeed quite generally, cf. [Lau-CC]). Tautologically we have χc (A1 /k, j0 ! N ) = χc (Gm /k, N ), and again χc (Gm /k, N ) = χ(Gm /k, N ). 5. For any perverse sheaf N on Gm /k, whether or not in P, the groups Hci (Gm /k, N ) vanish for i < 0 and for i > 1. Proof. Using the long exact cohomology sequence, we reduce immediately to the case when N is irreducible. If N is punctual, the assertion 24 3. FIBRE FUNCTORS is obvious.

G → G I(∞) → 0, where G I(∞) is viewed as a punctual sheaf supported at ∞. We view this as a short exact sequence of perverse sheaves 0 → G I(∞) → j! G[1] = j∞ ! j0 ! G[1] → j∞ j0 ! G[1] → 0. Similarly, we have a short exact sequence of perverse sheaves 0 → j∞ j0 ! G[1] → Rj∞ j0 ! G[1] → H 1 (I(∞), G) → 0, where now H 1 (I(∞), G) is viewed as a punctual sheaf supported at ∞. Taking their cohomology sequences on P1 , we get short exact sequences 0 → G I(∞) → Hc0 (Gm /k, G[1]) → H 0 (P1 /k, j∞ j0 !

Download PDF sample

Rated 4.99 of 5 – based on 36 votes