Computing zeta functions of arithmetic schemes

Proc. Lond. Math. Soc. 111 (2015), no. 6, 1379–1401 (DOI).

arXiv preprint (September 2015).

Abstract

We present new algorithms for computing zeta functions of algebraic varieties over finite fields. In particular, let X be an arithmetic scheme (scheme of finite type over Z), and for a prime p let ζXp(s) be the local factor of its zeta function. We present an algorithm that computes ζXp(s) for a single prime p in time p1/2+o(1), and another algorithm that computes ζXp(s) for all primes p < N in time N (log N)3+o(1). These generalise previous results of the author from hyperelliptic curves to completely arbitrary varieties.


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