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Computing zeta functions of arithmetic schemes
Author(s) -
Harvey David
Publication year - 2015
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdv056
Subject(s) - mathematics , prime (order theory) , scheme (mathematics) , finite field , riemann zeta function , algebraic number , prime factor , arithmetic , function (biology) , discrete mathematics , arithmetic zeta function , prime zeta function , combinatorics , pure mathematics , mathematical analysis , evolutionary biology , biology
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 letζ X p( s )be the local factor of its zeta function. We present an algorithm that computesζ X p( s )for a single prime p in time p 1 / 2 + o ( 1 ) , and another algorithm that computesζ X p( s )for all primes p < N in time N log 3 + o ( 1 ) N . These generalise previous results of the author from hyperelliptic curves to completely arbitrary varieties.