Division by 2 on odd degree hyperelliptic curves and their jacobians (original) (raw)

Division by 2 on hyperelliptic curves and jacobians

2016

Let KKK be an algebraically closed field of characteristic different from 2, ggg a positive integer, f(x)f(x)f(x) a degree (2g+1)(2g+1)(2g+1) polynomial with coefficients in KKK and without multiple roots, C:y2=f(x)C: y^2=f(x)C:y2=f(x) the corresponding genus ggg hyperelliptic curve over KKK and JJJ the jacobian of CCC. We identify CCC with the image of its canonical embedding into JJJ (the infinite point of CCC goes to the zero point of JJJ). For each point P=(a,b)inC(K)P=(a,b)\in C(K)P=(a,b)inC(K) there are 22g2^{2g}22g points frac12PinJ(K)\frac{1}{2}P \in J(K)frac12PinJ(K). We describe explicitly the Mumford represesentations of all frac12P\frac{1}{2}Pfrac12P. The rationality questions for frac12P\frac{1}{2}Pfrac12P are also discussed.

Torsion points of small order on hyperelliptic curves

European Journal of Mathematics, 2022

Let C be a hyperelliptic curve of genus g > 1 over an algebraically closed field K of characteristic zero and O one of the (2g + 2) Weierstrass points in C(K). Let J be the jacobian of C, which is a g-dimensional abelian variety over K. Let us consider the canonical embedding of C into J that sends O to the zero of the group law on J. This embedding allows us to identify C(K) with a certain subset of the commutative group J(K). A special case of the famous theorem of Raynaud (Manin-Mumford conjecture) asserts that the set of torsion points in C(K) is finite. It is well known that the points of order 2 in C(K) are exactly the "remaining" (2g + 1) Weierstrass points. One of the authors [12] proved that there are no torsion points of order n in C(K) if 3 ≤ n ≤ 2g. So, it is natural to study torsion points of order 2g + 1 (notice that the number of such points in C(K) is always even). Recently, the authors proved that there are infinitely many (for a given g) mutually nonisomorphic pairs (C, O) such that C(K) contains at least four points of order 2g + 1. In the present paper we prove that (for a given g) there are at most finitely many (up to a isomorphism) pairs (C, O) such that C(K) contains at least six points of order 2g + 1.

Characterization of the torsion of the Jacobians of two families of hyperelliptic curves

Acta Arithmetica 12/2013; 161(3):201-218.

Consider the families of curves Cn,A:y2=x^n+Ax and Cn,A:y2=x^n+A where A is a nonzero rational. Let Jn,A and Jn,A denote their respective Jacobian varieties. The torsion points of C3,A(Q) and C3,A(Q) are well known. We show that for any nonzero rational A the torsion subgroup of J7,A(Q) is a 2-group, and for A<>4a^4,−1728,−1259712 this subgroup is equal to J7,A(Q)[2] (for a excluded values of A, with the possible exception of A=−1728, this group has a point of order 4). This is a variant of the corresponding results for J3,A (A<>4) and J5,A. We also almost completely determine the Q-rational torsion of Jp,A for all odd primes p, and all A∈Q∖{0}. We discuss the excluded case (i.e. A∈(−1)^(p−1)/2*pN^2).

Halves of Points of an Odd Degree Hyperelliptic Curve in its Jacobian

Integrable Systems and Algebraic Geometry, 2020

Let f (x) be a degree (2g +1) monic polynomial with coefficients in an algebraically closed field K with char(K) = 2 and without repeated roots. Let R ⊂ K be the (2g + 1)-element set of roots of f (x). Let C : y 2 = f (x) be an odd degree genus g hyperelliptic curve over K. Let J be the jacobian of C and J[2] ⊂ J(K) the (sub)group of points of order dividing 2. We identify C with the image of its canonical embedding into J (the infinite point of C goes to the identity element of J).

Torsion Points of order 2g+1 on odd degree hyperelliptic curves of genus g

arXiv (Cornell University), 2019

Let K be an algebraically closed field of characteristic different from 2, g a positive integer, f (x) ∈ K[x] a degree 2g + 1 monic polynomial without multiple roots, C f : y 2 = f (x) the corresponding genus g hyperelliptic curve over K, and J the Jacobian of C f. We identify C f with the image of its canonical embedding into J (the infinite point of C f goes to the zero of the group law on J). It is known [9] that if g ≥ 2, then C f (K) contains no points of orders lying between 3 and 2g. In this paper we study torsion points of order 2g + 1 on C f (K). Despite the striking difference between the cases of g = 1 and g ≥ 2, some of our results may be viewed as a generalization of well-known results about points of order 3 on elliptic curves. E.g., if p = 2g + 1 is a prime that coincides with char(K), then every odd degree genus g hyperelliptic curve contains at most two points of order p. If g is odd and f (x) has real coefficients, then there are at most two real points of order 2g + 1 on C f. If f (x) has rational coefficients and g ≤ 51, then there are at most two rational points of order 2g + 1 on C f. (However, there exist odd degree genus 52 hyperelliptic curves over Q that have at least four rational points of order 105.

Computational Aspects of Hyperelliptic Curves

We introduce a new approach of computing the automorphism group and the field of moduli of points p = [C] in the moduli space of hyperelliptic curves Hg. Further, we show that for every moduli point p ∈ Hg(L) such that the reduced automorphism group of p has at least two involutions, there exists a representative C of the isomorphism class p which is defined over L.

Hyperelliptic Curves With Given A-number

Arxiv preprint math.NT/0401008, 2004

In this paper, we show that there exist families of curves (defined over an algebraically closed field k of characteristic p) whose Jacobians have interesting ptorsion. For example, we produce families of curves of large dimension so that the p-torsion of the Jacobian of each fibre contains multiple copies of the group scheme α p. The method is to study curves which admit a (Z/2) n-cover of the projective line. As a result, some of these families intersect the hyperelliptic locus H g. For small values of the a-number these families of curves intersect the hyperelliptic locus H g , resulting in the following corollaries. Corollary 1.2. Suppose g ≥ 2 and p ≥ 5. There exists a (g − 2)-dimensional family of smooth hyperelliptic curves of genus g all of whose fibres have a-number at least 2. Corollary 1.3. Suppose g ≥ 3 is odd and p ≥ 7. There exists a (g − 5)/2-dimensional family of smooth hyperelliptic curves of genus g whose fibres have a-number at least 3. Consider the group scheme k[F,V ]/(F 3 ,V 3 , F 2 +V 2). This group scheme corresponds to the p-torsion of an abelian variety of dimension 2 which has a-number 1 and prank 0. Corollary 1.4. For all g ≥ 4, there exists a (g − 2)-dimensional family of smooth curves X in H g,2 so that Jac(X)[p] ≃ k[F,V ]/(F 3 ,V 3 , F 2 +V 2) ⊕ (µ p ⊕ Z/p) g−2. In fact, we show that when k ≥ 2 and 2 k ≤ g < 2 k + 2 k−1 then there exists a hyperelliptic curve X with Jac(X)[p] ≃ k[F,V ]/(F 3 ,V 3 , F 2 +V 2) ⊕ (µ p ⊕ Z/p) g−2. Our method is to analyze the curves in the locus H g,n in terms of fibre products of hyperelliptic curves. We extend results of Kani and Rosen [6] to compare the p-torsion of the Jacobian of a curve X in H g,n to the p-torsion of the Jacobians of its Z/2Z-quotients up to isomorphism. In particular, we use Yui's description of the p-torsion of the Jacobian of a hyperelliptic curve in terms of the branch locus [15]. In some cases, this reduces the study of the p-torsion of the Jacobian of X to the study of the intersection of some subvarieties in the configuration space of branch points. The results above on families of curves having prescribed group schemes in the ptorsion of their Jacobians are all found in Section 4. In Section 2, we describe properties of the locus H g,n which are used in Section 4, including the proof that a curve X in this locus is the fibre product of n hyperelliptic curves and the comparison of the Jacobian of X with the Jacobians of its quotients. As an unrelated geometric result, we show that H g,2 is connected if and only if g ≤ 3 in Theorem 2.13. In Section 3, we study the the branch loci corresponding to non-ordinary hyperelliptic curves and prove some intersection results for this subvariety of the configuration space which are again used in Section 4. The limitation of the approach is to understand the p-torsion of Jacobians of hyperelliptic curves, especially in terms of their branch loci. Any progress in this direction can be used to extend these results. For example, we use recent work of [4] to prove Corollary 4.13 on the dimension of the intersection of H g,2 with the locus of curves of genus g having prank exactly f. We would like to thank E. Goren for suggesting the topic of this paper and E. Kani for help with Proposition 2.5. 2 Fibre products of hyperelliptic curves Let k be an algebraically closed field of characteristic p = 2. Let G be an elementary abelian 2-group of order 2 n. In this section, we describe G-Galois covers f : X → P 1 k