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Point counting on curves using a gonality preserving lift. (English) Zbl 1442.14096

Summary: We study the problem of lifting curves from finite fields to number fields in a genus and gonality preserving way. More precisely, we sketch how this can be done efficiently for curves of gonality at most four, with an in-depth treatment of curves of genus at most five over finite fields of odd characteristic, including an implementation in Magma. We then use such a lift as input to an algorithm due to the second author for computing zeta functions of curves over finite fields using \(p\)-adic cohomology.

MSC:

14H25 Arithmetic ground fields for curves
11G05 Elliptic curves over global fields
14F30 \(p\)-adic cohomology, crystalline cohomology
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14Q05 Computational aspects of algebraic curves
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14G05 Rational points
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