Rational torsion of J0(N) for hyperelliptic modular curves and families of Jacobians of genus 2 and genus 3 curves with a rational point of order 5,7 or 10 (original) (raw)

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).

On the torsion of the Jacobians of the hyperelliptic curves y^2=x^n+a and y^2=x(x^n+a)

Consider two families of hyperelliptic curves (over Q) C^n,a : y^2 = x^n + a and C_n,a : y^2 = x(x^n + a), and their Jacobians J^n,a, J_n,a respectively. We give the partial characterization of the torsion part of J^n,a (Q) and J_n,a (Q). More precisely, we show that the only prime factors of the orders of such groups are 2 and prime divisors of n (we also give the upper bounds for the exponents). Moreover, we give the complete description of the torsion part of J_8,a (Q). Namely, we show that J_8,a (Q)tors = J_8,a (Q) [2]. In addition, we characterize the torsion parts of J_p,a (Q), where p is an odd prime, and of J^n,a (Q), where n = 4, 6, 8. The main ingredients in the proofs are explicit computations of zeta functions of the title curves in some cases, which are of independent interest, and applications of the Chebotarev Density Theorem.

University of Groningen Modular invariants for genus 3 hyperelliptic curves

2018

In this article we prove an analogue of a theorem of Lachaud, Ritzenthaler, and Zykin, which allows us to connect invariants of binary octics to Siegel modular forms of genus 3. We use this connection to show that certain modular functions, when restricted to the hyperelliptic locus, assume values whose denominators are products of powers of primes of bad reduction for the associated hyperelliptic curves. We illustrate our theorem with explicit computations. This work is motivated by the study of the values of these modular functions at CM points of the Siegel upper half-space, which, if their denominators are known, can be used to effectively compute models of (hyperelliptic, in our case) curves with CM.

Modular invariants for genus 3 hyperelliptic curves

Research in Number Theory

Constructing genus-3 hyperelliptic Jacobians with CM

LMS Journal of Computation and Mathematics

Given a sextic CM field$K$, we give an explicit method for finding all genus-$3$hyperelliptic curves defined over$\mathbb{C}$whose Jacobians are simple and have complex multiplication by the maximal order of this field, via an approximation of their Rosenhain invariants. Building on the work of Weng [J. Ramanujan Math. Soc.16 (2001) no. 4, 339–372], we give an algorithm which works in complete generality, for any CM sextic field$K$, and computes minimal polynomials of the Rosenhain invariants for any period matrix of the Jacobian. This algorithm can be used to generate genus-3 hyperelliptic curves over a finite field mathbbFp\mathbb{F}_{p}mathbbFpwith a given zeta function by finding roots of the Rosenhain minimal polynomials modulo ppp.

Division by 2 on odd degree hyperelliptic curves and their jacobians

Izvestiya: Mathematics, 2019

Let be an algebraically closed field of characteristic different from , a positive integer, a polynomial of degree with coefficients in and without multiple roots, the corresponding hyperelliptic curve of genus over , and its Jacobian. We identify with the image of its canonical embedding in (the infinite point of goes to the identity element of ). It is well known that for every there are exactly elements such that . Stoll constructed an algorithm that provides the Mumford representations of all such in terms of the Mumford representation of . The aim of this paper is to give explicit formulae for the Mumford representations of all such in terms of the coordinates , where is given by a point . We also prove that if , then does not contain torsion points of orders between and .

Non-hyperelliptic modular curves of genus 3

Journal of Number Theory, 2010

A curve C defined over Q is modular of level N if there exists a nonconstant morphism X1(N) −→ C defined over Q for some positive integer N. We present an algorithm to compute explicitly equations for modular non-hyperelliptic curves defined over Q of genus 3. Let C be a modular curve of level N , we say that C is new if the corresponding morphism between J1(N) and Jac(C) factorizes through the new part of J1(N). We compute equations of 44 non-hyperelliptic new modular curves of genus 3, that we conjecture to be the complete list of this kind of curves. Furthermore, we describe some aspects of non-new modular curves.

Modular curves and the eisenstein ideal

Publications mathématiques de l'IHÉS, 1977

Modular curves and the Eisenstein ideal Publications mathématiques de l'I.H.É.S., tome 47 (1977), p. 33-186 http://www.numdam.org/item?id=PMIHES\_1977\_\_47\_\_33\_0 © Publications mathématiques de l'I.H.É.S., 1977, tous droits réservés. L'accès aux archives de la revue « Publications mathématiques de l'I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ MODULAR CURVES AND THE EISENSTEIN IDEAL by B. MAZUR (1) This result may be used to provide a complete determination of the possible torsion subgroups of Mordell-Weil groups of elliptic curves over Q. Namely: Theorem (8).-Let 0 be the torsion subgroup of the Mordell-Weil group of an elliptic curve defined over Q^. Then 0 is isomorphic to one of the following 15 groups: Z/TTI.Z for m^io or m==i2 or: (Z/2.Z)x(Z/2v.Z) for v^4. (III? (5-1)-Sy [27] theorems 7 and 8 are implied by theorem 7 for prime values of 772^23. See also the discussion of this problem in [49].

A note on the torsion of the Jacobians of superelliptic curves y^{q}=x^{p}+a

This article is a short version of the paper published in J. Number Theory 145 (2014) but we add new results and a brief discussion about the Torsion Conjecture. Consider the family of superelliptic curves (over Q) Cq,p,a : y q = x p + a, and its Jacobians Jq,p,a, where 2 < q < p are primes. We give the full (resp. partial) characterization of the torsion part of J3,5,a (Q) (resp. Jq,p,a (Q)). The main tools are computations of the zeta function of C 3,5,a (resp. C q,p,a) over F l for primes l ≡ 1, 2, 4, 8, 11(mod 15) (resp. for primes l ≡ −1(mod qp)) and applications of the Chebotarev Density Theorem.

Rational torsion of J0(N) for hyperelliptic modular curves and families of Jacobians of genus 2 and genus 3 curves with a rational point of order 5,7 or 10 (original) (raw)

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