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Perversity and exponential sums. (English) Zbl 0755.14008

Algebraic number theory - in honor of K. Iwasawa, Proc. Workshop Iwasawa Theory Spec. Values \(L\)-Funct., Berkeley/CA (USA) 1987, Adv. Stud. Pure Math. 17, 209-259 (1989).
[For the entire collection see Zbl 0721.00006.]
Let \(X\) be a closed subscheme of the \(r\)-dimensional affine space \(\mathbb{A}^ r\) over \(\mathbb{Z}\), such that it is irreducible and reduced over \(\mathbb{C}\), of dimension \(n\), say. One defines a nonnegative integer invariant \(A\) of \(X\), namely the difference of the Euler-PoincarĂ© characteristics of \(X_ \mathbb{C}\) and of a general hyperplane section, with coefficients in the intersection complex of \(X_ \mathbb{C}\). — For a fixed prime \(p\) one takes the normalized (i.e., divided by \(p^{n/2})\) characteristic function of \(X(\mathbb{F}_ p)\) on \(\mathbb{A}^ r(\mathbb{F}_ p)\). Its Fourier transform is a complex-valued exponential sum \(f_ p\) on \((\mathbb{F}_ p)^ r\). One obtains bounds for the \(L^ 1\)-norm of \(f_ p\) and its variance (here \(A\) occurs), thereby distinguishing the cases \(A=0\), \(A=1\), \(A\geq 2\). The invariant \(A\) is determined for special choices of \(X\), e.g. of hypersurfaces.
For the proofs, the situation is generalized to that with a given \(\ell\)- adic sheaf \({\mathcal F}\) on \(X[1/\ell]\) and one uses \(\ell\)-adic Fourier- transformation. To cope with singular \(X\), one uses the theory of perversity and passes to the middle extension of \({\mathcal F}\) from the nonsingular part of \(X\).

MSC:

14G15 Finite ground fields in algebraic geometry
11T23 Exponential sums
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)

Citations:

Zbl 0721.00006
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