Quantum computation of zeta functions of curves

Authors:
Kiran S. Kedlaya
Affiliations:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA
Venue:
Computational Complexity
Year:
2006

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Abstract

We exhibit a quantum algorithm for determining the zeta function of a genus g curve over a finite field Fq, which is polynomial in g and log(q) This amounts to giving an algorithm to produce provably random elements of the class group of a curve, plus a recipe for recovering a Well polynomial from enough of its cyclic resultants. The latter effectivizes a result of Fried in a restricted setting.