By Nicholas M. Katz
Convolution and Equidistribution explores a tremendous element of quantity theory--the thought of exponential sums over finite fields and their Mellin transforms--from a brand new, express perspective. The ebook offers essentially vital effects and a plethora of examples, beginning up new instructions within the topic. The finite-field Mellin rework (of a functionality at the multiplicative workforce of a finite box) is outlined via summing that functionality opposed to variable multiplicative characters. the elemental query thought of within the booklet is how the values of the Mellin rework are allotted (in a probabilistic sense), in instances the place the enter functionality is definitely algebro-geometric. this query is responded via the book's major theorem, utilizing a mix of geometric, specific, and group-theoretic equipment. via supplying a brand new framework for learning Mellin transforms over finite fields, this publication opens up a brand new means for researchers to extra discover the topic.
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Extra resources for Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms
Also the objects which are both ι-pure of weight zero and arithmetically semisimple form a full Tannakian subcategory Parith,ι wt=0,ss . Pick one of the two possible isomorphisms of G/k with Gm /k, viewed as A1 /k \ 0. Denote by j0 : Gm /k ⊂ A1 /k the inclusion, and denote by ω the fibre functor on Parith defined by N → ω(N ) := H 0 (A1 /k, j0 ! N ). The definition of this ω depends upon the choice of one of the two kpoints at infinity on the complete nonsingular model, call it X, of G/k. If G/k is already Gm /k, then these two points at infinity on X are both k-rational.
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. , Hc−1 (Gm /k, F) = Hc0 (Gm /k, F) = 0; the groups Hci (Gm /k, F) = Hci+1 (Gm /k, F) with i ≤ −2 or i ≥ 2 vanish trivially.
1 (i)]. 1. Suppose that the tensor product sheaf G := F ⊗ [x → 1/x] F is itself a middle extension sheaf; this is automatic if either F is lisse on G or if the finite set S of points of G(k) at which F is ramified is disjoint from the set 1/S of its inverses. Suppose further that G is 53 54 10. A FIRST CONSTRUCTION OF AUTODUAL OBJECTS geometrically irreducible, and not geometrically isomorphic to an Lχ . Then G(1/2) is a geometrically irreducible object of Parith which is symplectically self-dual.
Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms by Nicholas M. Katz