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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.

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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