A Recursive Method for Computing Zeta Functions of Varieties

Abstract

This paper is dedicated to Richard P. Brent on the occasion of his sixtieth birthday. AbstractWe present an algorithm that reduces the problem of calculating a numerical approximation to the action of absolute Frobenius on the middle-dimensional rigid cohomology of a smooth projective variety over a finite field, to that of performing the same calculation for a smooth hyperplane section. When combined with standard geometric techniques, this yields a method for computing zeta functions which proceeds 'by induction on the dimension'. The 'inductive step' combines previous work of the author on the deformation of Frobenius with a higher rank generalisation of Kedlaya's algorithm. The analysis of the loss of precision during the algorithm uses a deep theorem of Christol and Dwork on p-adic solutions to differential systems at regular singular points. We apply our algorithm to compute the zeta functions of compactifications of certain surfaces which are double covers of the affine plane.

Ranks of Elliptic Curves Over Function Fields

“…The truth of this conjecture immediately yields improvements to the running time of the original fibration method, see [19,Examples 9.2,9.3], and also the refined method of this paper. We prove the conjecture, under modest hypotheses (Theorem 8.4).…”

“…Second, in [19,Section 9.3] some new ideas on how to significantly improve the practical performance of the fibration method are sketched, although no details are offered. We work out these ideas in great detail and use them in our calculations for elliptic curves.…”

“…In an earlier paper, the present author introduced a new method for computing zeta functions of varieties over finite fields [19]. The principal novelty of the method was that it proceeded by induction on the dimension of the variety.…”

Point counting on K3 surfaces and an application concerning real and complex multiplication

“…There is at least one further idea, which we can mention only in passing, namely the use of a fibration. For more information, we advise the reader to consult the articles [29] of A. Lauder and [34] of S. Pancratz and J. Tuitman.…”

Point Counting in Families of Hyperelliptic Curves

“…This lemma is from [19], but we give a slightly different proof. Truncating the right hand side in the computation ofC i modulo p Na in step 3 implies thatC is a solution of rĊ +CH = p Na E with E an integral matrix.…”