Counting hyperelliptic curves

Abstract. We introduce several new methods to obtain upper bounds on the number of solutions of the congruenceswith a prime p and a polynomial f , where (x, y) belongs to an arbitrary square with side length M . We use these results and methods to derive non-trivial upper bounds for the number of hyperelliptic curvesover the finite field F p of p elements, with coefficients in a 2g-dimensional cubethat are isomorphic to a given curve and give an almost sharp lower bound on the number of non-isomorphic hyperelliptic curves with coefficients in that cube. Furthermore, we study the size of the smallest box that contain a partial trajectory of a polynomial dynamical system over F p .

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“…By (29), (30), (31), the absolute value of the both hand side is bounded by pM 1+o(1) T −1 . Thus, we get…”

Section: 1mentioning

confidence: 99%

“…It is known (see [22,29]) that the number of non isomorphic hyperelliptic curves of genus g over F p is 2p 2g−1 + O(gp 2g−2 ). We address here the problem of estimating from below, the number of non-isomorphic hyperelliptic curves of genus g over F p , H a , when a = (a 0 , .…”

mentioning

confidence: 99%

Points on curves in small boxes and applications

Chang¹,

Cilleruelo²,

Garaev³

et al. 2014

Michigan Math. J.

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“…Since we are only interested in curves up to birational equivalence, rather than simply enumerating all polynomials of a given form one could instead enumerate curves by their moduli. Questions of this type in low genus have been pursued by many authors: Cardona, Nart, and Pujolàs [4] and Espinosa García, Hernández Encinas, and Muñoz Masqué [8] study genus 2; Nart and Sadornil [12] study hyperelliptic curves of genus 3; Nart and Ritzenthaler [11] study nonhyperelliptic curves of genus 3 over fields of even characteristic; and Nart [10] gives a closed formula for the number of hyperelliptic curves in odd characteristic. In this paper we used a more naive approach since it is more transparent, easier to implement, and at the same time still feasible.…”

Section: Computational Resultsmentioning

confidence: 99%

Nondegenerate curves of low genus over small finite fields

Castryck¹,

Voight²

2010

Arithmetic, Geometry, Cryptography and Coding Theory 2009

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In a previous paper, we proved that over a finite field k of sufficiently large cardinality, all curves of genus at most 3 over k can be modeled by a bivariate Laurent polynomial that is nondegenerate with respect to its Newton polytope. In this paper, we prove that there are exactly two curves of genus at most 3 over a finite field that are not nondegenerate, one over F 2 and one over F 3. Both of these curves have extremal properties concerning the number of rational points over various extension fields. 1991 Mathematics Subject Classification. Primary 14H45, Secondary 14M25. Key words and phrases. nondegenerate curves, finite fields, Newton polytope. The first author is a postdoctoral fellow of FWO-Vlaanderen. He would like to thank Alessandra Rigato for some helpful comments on curves over finite fields having many or few rational points.

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“…For instance, this technique has been used in the papers [3] and [11] to count, for any given value of the genus g > 1, the number of k-isomorphism classes of hyperelliptic curves, of pointed hyperelliptic curves, and of hyperelliptic curves having a rational Weierstrass point. N = 1, 2 The description of the subtypes of conjugacy classes of PGL N +1 (k) and the computation of the coefficients c α can be deduced from [7, Proposition 2.3, Lemma 2.4] for N = 1 and from [9, for N = 2.…”

Section: Theorem 34 For a Fixed Value Of N 1 The Generating Functimentioning

confidence: 99%