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.
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Additional resources for Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms
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 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 = j∞ ! j0 ! G → j∞ j0 ! G → 0. Similarly, we have a short exact sequence of perverse sheaves 0 → j∞ j0 ! G → Rj∞ j0 ! G → 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) → H 0 (P1 /k, j∞ j0 !