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Exponential sums and Newton polyhedra: cohomology and estimates. (English) Zbl 0723.14017
In this important paper the authors deal with exponential sums on varieties $$V$$ of the form: $$V=({\mathbb C}_ m)^ r\times {\mathbb A}^ s (r+s=n)$$. Namely, for a function $$f\in {\mathbb F}_ q[x_ 1,...,x_ n,(x_ 1...x_ r)^{-1}]$$, and a nontrivial additive character $$\Psi$$ of $${\mathbb F}_ q$$, $$q=p^ a$$, $$p=\text{char}({\mathbb F}_ q)$$ they consider the sums $$S_ i=\sum_{x\in V({\mathbb F}_{q^ i})}\Psi \circ \text{Tr}_{{\mathbb F}_{q^ i}/\mathbb F_ q}(f(x))$$ and the corresponding $$L$$-function: $$L(t)=\exp \Bigl(\sum^{\infty}_{i=1}S_ it^ i/i\Bigr)$$. – Sums over more general varieties can be reduced to this situation.
In the style of Dwork’s study of the $$p$$-adic cohomology of smooth hypersurfaces of characteristic $$p$$ the authors associate to $$V$$ and $$f$$ a complex of $$p$$-adic Banach spaces on which the Frobenius operates. This is constructed in such a way that the Dwork trace formula expresses $$L(t)$$ as an alternate product of characteristic polynomials of Frobenius operating in cohomology. In previous papers, the authors were able to relate certain properties of $$L(t)$$ to combinatorial invariants of the Newton polyhedron $$\Delta (f)$$ of $$f$$. In particular, degree and total degree of $$L(t)$$ were bounded in terms of certain Euclidean volumes related to $$\Delta(f)$$. The main technical point of the present paper consists in proving that under non-degeneracy conditions on $$f$$, the afore-mentioned complex is acyclic in non-zero dimensions. The proof is based on ring-theoretic results of Kouchnirenko and Hochster. As a consequence the previous estimates on the degrees $$L(t)$$ are shown to be sharp. Under nondegeneracy assumptions, a combinatorial lower bound for the Newton polygon of $$L(t)$$ is also obtained.
Finally, the previous $$p$$-adic results are used to determine the dimension of the $$\ell$$-adic cohomology spaces $$H^ i_ C({\mathbb C}^ n_ m\times {\bar {\mathbb F}}_ q,{\mathcal L}_{\Psi}(f))$$ and conditions for $$H^ n_ C({\mathbb G}^ n_ m\times {\bar {\mathbb F}}_ q,{\mathcal L}_{\Psi}(f))$$ to be pure of weight $$n$$. From this, via Deligne’s results, the authors obtain archimedean estimates for the roots of $$L(t)$$. This is applied in particular to sums over the affine space: these results generalize a theorem of Deligne on the structure of the $$L$$-function when the leading form of $$f$$ defines a nonsingular projective hypersurface. In a section dedicated to examples, the authors reestablish the link with Dwork’s classical results on the zeta function of a regular hypersurface.

MSC:
 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 11T23 Exponential sums 14G15 Finite ground fields in algebraic geometry 11L40 Estimates on character sums 14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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