Publisher Correction: Enumerating odd-degree hyperelliptic curves and abelian surfaces over $${\mathbb {P}}^1$$
نویسندگان
چکیده
منابع مشابه
Most Odd Degree Hyperelliptic Curves Have Only One Rational Point
Consider the smooth projective models C of curves y = f(x) with f(x) ∈ Z[x] monic and separable of degree 2g + 1. We prove that for g ≥ 3, a positive fraction of these have only one rational point, the point at infinity. We prove a lower bound on this fraction that tends to 1 as g → ∞. Finally, we show that C(Q) can be algorithmically computed for such a fraction of the curves. The method can b...
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We enumerate the singular algebraic curves in a complete linear system on a smooth projective surface. The system must be suitably ample in a rather precise sense. The curves may have up to eight nodes, or a triple point of a given type and up to three nodes. The curves must also pass through appropriately many general points. The number of curves is given by a universal polynomial in four basi...
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Consider the smooth projective models C of curves y = f(x) with f(x) ∈ Z[x] monic and separable of degree 2g + 1. We prove that for g ≥ 3, a positive fraction of these have only one rational point, the point at infinity. We prove a lower bound on this fraction that tends to 1 as g →∞. Finally, we show that C(Q) can be algorithmically computed for such a fraction of the curves, via Chabauty’s me...
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In this paper we propose a very simple and efficient encoding function from Fq to points of a hyperelliptic curve over Fq of the form H : y = f(x) where f is an odd polynomial. Hyperelliptic curves of this type have been frequently considered in the literature to obtain Jacobians of good order and pairing-friendly curves. Our new encoding is nearly a bijection to the set of Fq-rational points o...
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We prove that if F is an infinite field with characteristic different from 2, whose theory is supersimple, and C is an elliptic or hyperelliptic curve over F with generic moduli then C has a generic F -rational point. The notion of generity here is in the sense of the supersimple field F .
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ژورنال
عنوان ژورنال: Mathematische Zeitschrift
سال: 2023
ISSN: ['1432-1823', '0025-5874']
DOI: https://doi.org/10.1007/s00209-023-03283-w