A Recursive Method for Computing Zeta Functions of Varieties

We present a method for calculating the zeta function of a smooth projective variety over a finite field which proceeds by induction on the dimension. Specifically, we outline an algorithm which reduces the problem of calculating a numerical approximation for the action of Frobenius on the middle-dimensional rigid cohomology of a smooth variety, to that of performing the same calculation for a smooth hyperplane section. We present in detail the main new algorithmic ingredient under some simplifying assumptions, and give full details of our algorithm for calculating zeta functions for some specific surfaces; we call it the “fibration algorithm”. We have implemented the fibration algorithm for these surfaces over prime fields using the Magma programming language, and present some explicit examples which we have computed. To illustrate the main idea behind our approach, we begin by outlining the proof given by Deligne of the Riemann hypothesis for a smooth projective variety X over the finite field Fq [9]. Specifically, the statement that for each 0 ≤ i ≤ 2 dim(X) the action of the Frobenius endomorphism on the `-adic etale cohomology space H et(X,Q`) has eigenvalues of complex absolute value q. Let X ⊂ P be a smooth projective variety of dimension n + 1 > 1 defined over the finite field Fq. Denote by P the dual projective space whose points t correspond to hyperplanes Ht in P, and let D be a line in P. Let X ⊂ X ×D denote the set of points (x, t) such that x ∈ Ht. Projection on the first and second coordinates yields maps X π ← X f → D. The fibre of f at t ∈ D is the hyperplane section Xt = X ∩ Ht of X. For sufficiently general D these maps define a Lefschetz pencil [9, (5.1)] (one may need to change the projective embedding first [9, (5.7)]). The action of the Frobenius endomorphism on the `-adic etale cohomology H∗ et(X,Q`) may be studied via this Lefschetz pencil. In particular, assuming the result holds for smooth curves and arguing by induction on the dimension n+1,

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