Castryck, Wouter; Tuitman, Jan Point counting on curves using a gonality preserving lift. (English) Zbl 1442.14096 Q. J. Math. 69, No. 1, 33-74 (2018). 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. Cited in 4 Documents 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 PDFBibTeX XMLCite \textit{W. Castryck} and \textit{J. Tuitman}, Q. J. Math. 69, No. 1, 33--74 (2018; Zbl 1442.14096) Full Text: DOI arXiv