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Genus-2 Jacobians with torsion points of large order. (English) Zbl 1319.14030
This short and clear article is concerned with the explicit construction of smooth curves of genus $$2$$ over $$\mathbb{Q}$$ whose Jacobian varieties admit a torsion point of large order. Among the curves constructed is one whose Jacobian admits a torsion point over $$\mathbb{Q}$$ of order $$70$$; moreover, the author obtains infinitely many curves whose Jacobians all have a torsion point of order $$48$$.
Two main methods are used to construct the curves in question. The first of these uses techniques developed in [R. Bröker et al., LMS J. Comput. Math. 18, 170–197 (2015; Zbl 1387.14086)]: given two elliptic curves $$E_1$$ and $$E_2$$ along with an isomorphism $$\psi : E_1  \to E_2 $$ that is an anti-isometry with respect to the Weil pairings, the quotient of $$E_1 \times E_2$$ by the graph of $$\psi$$ is a principally polarized abelian variety in a natural way, and if this is indeed the Jacobian of a smooth curve of genus $$2$$, then an equation of such a curve can be effectively determined. Using two elliptic curves over $$\mathbb{Q}$$ with torsion points of order $$7$$ and $$10$$, respectively, this construction yields a curve of genus $$2$$ whose Jacobian admits a torsion point of order $$70$$ over the same field.
The aforementioned infinite family is constructed by using $$2$$-torsion instead of $$3$$-torsion in the argument above. In this case, the corresponding “gluing techniques” were developed in [E. W. Howe et al., Forum Math. 12, No. 3, 315–364 (2000; Zbl 0983.11037)]. In his application, the author first fixes a curve $$E_1$$ with an $$8$$-torsion point and shows that it can be glued with all curves $$E_2$$ in a certain family that depend rationally on $$2$$ parameters and whose members all admit a $$6$$-torsion point. The gluing techniques then yield a family of curves genus $$2$$ along with a torsion point of order $$24$$ on their Jacobian, which is parametrized by a rational variety $$V$$ defined over $$\mathbb{Q}$$. The subtlety lies in showing that for infinitely many points $$v$$ in $$V (\mathbb{Q})$$ the torsion point on the corresponding Jacobian is actually the double of another $$\mathbb{Q}$$-rational point. In geometric terms, the author accomplishes this by showing that there exists a map of surfaces $$U \to V$$ defined over $$\mathbb{Q}$$ such that $$v$$ is such a double if and only if it lies in the image of $$U (\mathbb{Q})$$, and subsequently exhibiting a curve of genus $$1$$ on $$U$$ with infinitely many $$\mathbb{Q}$$-rational points.
The article concludes by giving new examples of curves of genus $$2$$ that allow a defining equation with small coefficients, yet whose Jacobians still admit a torsion point of large order. These were found by a computer search.

##### MSC:
 14G05 Rational points 11G30 Curves of arbitrary genus or genus $$\ne 1$$ over global fields 14H25 Arithmetic ground fields for curves 14H45 Special algebraic curves and curves of low genus
##### Keywords:
curves of genus 2; torsion points; rational points
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##### References:
  Bröker, Genus-2 curves and Jacobians with a given number of points, LMS J. Comput. Math.  J. Cremona Elliptic curve data http://homepages.warwick.ac.uk/staff/J.E.Cremona/ftp/data/INDEX.html  N. D. Elkies Curves of genus 2 over Q whose Jacobians are absolutely simple abelian surfaces with torsion points of high order http://www.math.harvard.edu/elkies/g2_tors.html  Flynn, Large rational torsion on abelian varieties, J. Number Theory 36 pp 257– (1990) · Zbl 0757.14025 · doi:10.1016/0022-314X(90)90089-A  Howe, Improved upper bounds for the number of points on curves over finite fields, Ann. Inst. Fourier (Grenoble) 53 pp 1677– (2003) · Zbl 1065.11043 · doi:10.5802/aif.1990  Howe, Large torsion subgroups of split Jacobians of curves of genus two or three, Forum Math. 12 pp 315– (2000) · Zbl 0983.11037 · doi:10.1515/form.2000.008  Kani, The number of curves of genus two with elliptic differentials, J. reine angew. Math. 485 pp 93– (1997) · Zbl 0867.11045  Kubert, Universal bounds on the torsion of elliptic curves, Proc. London Math. Soc. 33 ((3)) pp 193– (1976) · Zbl 0331.14010 · doi:10.1112/plms/s3-33.2.193  Leprévost, Famille de courbes de genre 2 munies d’une classe de diviseurs rationnels d’ordre 13, C. R. Acad. Sci. Paris Sér. I Math. 313 pp 451– (1991) · Zbl 0758.14016  Leprévost, Familles de courbes de genre 2 munies d’une classe de diviseurs rationnels d’ordre 15,17,19 ou 21, C. R. Acad. Sci. Paris Sér. I Math. 313 pp 771– (1991) · Zbl 0758.14017  Leprévost, Torsion sur des familles de courbes de genre g, Manuscripta Math. 75 pp 303– (1992) · Zbl 0790.14021 · doi:10.1007/BF02567087  Leprévost, Jacobiennes de certaines courbes de genre 2: torsion et simplicité, J. Théor. Nombres Bordeaux 7 pp 283– (1995) · Zbl 0864.14017 · doi:10.5802/jtnb.144  Leprévost, Sur une conjecture sur les points de torsion rationnels des jacobiennes de courbes, J. reine angew. Math. 473 pp 59– (1996) · Zbl 0924.14015  Mazur, Rational points on modular curves, in: Modular functions of one variable, V pp 107– (1977) · Zbl 0357.14005 · doi:10.1007/BFb0063947  Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. 47 pp 33– (1977/78) · Zbl 0394.14008 · doi:10.1007/BF02684339  Mazur, Rational isogenies of prime degree (with an appendix by D. Goldfeld), Invent. Math. 44 pp 129– (1978) · Zbl 0386.14009 · doi:10.1007/BF01390348  Milne, Abelian varieties, in: Arithmetic geometry, Storrs, Conn. 1984 pp 103– (1986)  Ogawa, Curves of genus 2 with a rational torsion divisor of order 23, Proc. Japan Acad. Ser. A Math. Sci. 70 pp 295– (1994) · Zbl 0838.14021 · doi:10.3792/pjaa.70.295  Platonov, New orders of torsion points in Jacobians of curves of genus 2 over the rational number field, Dokl. Math. 85 pp 286– (2012) · Zbl 1319.11041 · doi:10.1134/S1064562412020330  Platonov, On the torsion problem in Jacobians of curves of genus 2 over the rational number field, Dokl. Math. 86 pp 642– (2012) · Zbl 1345.11045 · doi:10.1134/S1064562412050304  Platonov, On the simplicity of Jacobians for curves of genus 2 over the rational number field containing torsion points of large orders, Dokl. Math. 87 pp 318– (2013) · Zbl 1278.14045 · doi:10.1134/S1064562413030216  Stein, A database of elliptic curves-first report, in: Algorithmic number theory, Sydney, 2002 pp 267– (2002)
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