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Quickly constructing curves of genus 4 with many points. (English) Zbl 1417.11122
Kohel, David (ed.) et al., Frobenius distributions: Lang-Trotter and Sato-Tate conjectures. Winter school and workshop on Frobenius distributions on curves, CIRM, Marseille, France, February 17–21, 2014 and February 24–28, 2016. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 663, 149-173 (2016).
Summary: The defect of a curve over a finite field is the difference between the number of rational points on the curve and the Weil-Serre bound for the curve. We present a construction for producing genus-4 double covers of genus-2 curves over finite fields such that the defect of the double cover is not much more than the defect of the genus-2 curve. We give an algorithm that uses this construction to produce genus-4 curves with small defect. Heuristically, for all sufficiently large primes and for almost all prime powers q, the algorithm is expected to produce a genus-4 curve over $$\mathbb F_q$$ with defect at most 4 in time $$\widetilde{O}(q^{3/4})$$.
As part of the analysis of the algorithm, we present a reinterpretation of results of T. Hayashida [J. Math. Soc. Japan 20, 26–43 (1968; Zbl 0186.26501)] on the number of genus-2 curves whose Jacobians are isomorphic to the square of a given elliptic curve with complex multiplication by a maximal order. We show that a category of principal polarizations on the square of such an elliptic curve is equivalent to a category of right ideals in a certain quaternion order.
For the entire collection see [Zbl 1345.11004].

##### MSC:
 11G20 Curves over finite and local fields 14G05 Rational points 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 14G15 Finite ground fields in algebraic geometry
##### Keywords:
Jacobian; defect; rational points
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