# 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 ζ_{Xp}(*s*) be the local factor of its zeta function. We present an algorithm that computes ζ_{Xp}(*s*) for a single prime *p* in time *p*^{1/2+o(1)}, and another algorithm that computes ζ_{Xp}(*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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