By Nicholas M. Katz

ISBN-10: 0691153310

ISBN-13: 9780691153315

Convolution and Equidistribution explores an enormous element of quantity theory--the conception of exponential sums over finite fields and their Mellin transforms--from a brand new, specific viewpoint. The ebook provides essentially vital effects and a plethora of examples, starting up new instructions within the topic. The finite-field Mellin remodel (of a functionality at the multiplicative team of a finite box) is outlined by means of summing that functionality opposed to variable multiplicative characters. the fundamental query thought of within the publication is how the values of the Mellin rework are disbursed (in a probabilistic sense), in circumstances the place the enter functionality is certainly algebro-geometric. this question is responded by means of the book's major theorem, utilizing a mix of geometric, specific, and group-theoretic tools. through offering a brand new framework for learning Mellin transforms over finite fields, this publication opens up a brand new method 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**

**Example text**

Hence N itself is punctual. 4. , ι-pure of weight zero for every ι). If Ggeom,N is finite, then N is punctual. Proof. 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.

42 7. THE MAIN THEOREM We will now show that as E/k runs over larger and larger extensions of any degree, we have (1/#Good(E, N )) Trace(Λ(θE,ρ )) = O(1/ #E). ρ∈Good(E,N ) For good ρ, the term Trace(Λ(θE,ρ )) is Trace(F robE |Hc0 (G/k, M ⊗ Lρ )). For any ρ, the cohomology groups Hci (G/k, M ⊗ Lρ ) vanish for i = 0, cf. 1, so the Lefschetz Trace formula [Gr-Rat] gives Trace(F robE |Hc0 (G/k, M ⊗ Lρ )) = ρ(s)Trace(F robE,s |M ). 1] of Deligne’s Weil II, Hc0 (G/k, M ⊗ Lρ ) is ι-mixed of weight ≤ 0, so we have the estimate |Trace(F robE |Hc0 (G/k, M ⊗ Lρ ))| ≤ “ dim ”(M ).

It induces an autoduality on ω(N ) which is respected by Garith,N . Up to a scalar factor, this is the unique autoduality on ω(N ) which is respected by Garith,N , so it is either an orthogonal or a symplectic autoduality. We say that the duality has the sign +1 if it is orthogonal, and the sign −1 if it is symplectic. 1. Suppose that N in Parith is geometrically irreducible, ι-pure of weight zero, and arithmetically self-dual. Denote by the sign of its autoduality. For variable finite extension fields E/k, we have the estimate for | Trace((F rob2E,ρ |ω(N ))| = O(1/ #E).

### Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms by Nicholas M. Katz

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