## 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)

Zbl 0721.00006