Large torsion subgroups of split Jacobians of curves of genus two or three

We construct examples of families of curves of genus 2 or 3 over Q whose Jacobians split completely and have various large rational torsion subgroups. For example, the rational points on a certain elliptic surface over P^1 of positive rank parameterize a family of genus-2 curves over Q whose Jacobians each have 128 rational torsion points. Also, we find the genus-3 curve 15625(X^4 + Y^4 + Z^4) - 96914(X^2 Y^2 + X^2 Z^2 + Y^2 Z^2) = 0, whose Jacobian has 864 rational torsion points. This paper has appeared in Forum Math. 12 (2000) 315-364.

[1]  E. Kani.  The number of curves of genus two with elliptic differentials. , 1997 .

[2]  Everett W. Howe,et al.  Sous-groupes de torsion d'ordres élevés de jacobiennes décomposables de courbes de genre 2 , 1996 .

[3]  The Existence of Curves of Genus Two with Elliptic Differentials , 1997 .

[4]  R. Kuhn.  Curves of genus 2 with split Jacobian , 1988 .

[5]  F. Leprévost.  Points rationnels de torsion de jacobiennes de certains courbes de genre 2 , 1993 .

[6]  Everett W. Howe.  Constructing distinct curves with isomorphic Jacobians in characteristic zero , 1995 .

[7]  H. Baker.  Abelian Functions: Abel's Theorem and the Allied Theory of Theta Functions , 2008 .

[8]  Kenji Ueno,et al.  Principally polarized abelian variaties dimension two or three are Jacobian varieties , 1973 .

[9]  N. Katz,et al.  Arithmetic moduli of elliptic curves , 1985 .

[10]  Serge Lang,et al.  Abelian varieties , 1983 .

[11]  Hiroyuki Ogawa.  Curves of genus $2$ with a rational torsion divisor of order $23$ , 1994 .

[12]  E. V. Flynn.  Sequences of rational torsions on abelian varieties , 1991 .

[13]  S. Fermigier.  Exemples de courbes elliptiques de grand rang sur Q(t) et sur Q possédant des points d'ordre 2 , 1996 .

[14]  Jiri Dadok,et al.  Calibrations on ${\bf R}\sp 8$ , 1988 .

[15]  F. Leprévost.  Famille de courbes hyperelliptiques de genre g munies d'une classe de diviseurs rationnels d'ordre 2g^2+4g+1 , 1992 .

[16]  P. Deligne,et al.  Les Schémas de Modules de Courbes Elliptiques , 1973 .

[17]  Sur les fonctions zêta attachées aux classes de rayon , 1995 .

[18]  Jiri Dadok,et al.  Calibrations on R 8 , 1988 .

[19]  The Mordell-Weil Group of Curves of Genus 2 , 1983 .

[20]  Jacobiennes de certaines courbes de genre $2$ : torsion et simplicité , 1995 .

[21]  Y. Hasegawa.  Table of quotient curves of modular curves $X_0 \left( N \right)$ with genus $2$ , 1995 .

[22]  Y. Nakkajima.  On Infinitesimal Liftings and Degenerations of Hodge–de Rham Spectral Sequences , 1997 .

[23]  E. V. Flynn.  Large rational torsion on Abelian varieties , 1990 .

[24]  F. Leprévost.  Sur une conjecture sur les points de torsion rationnels des jacobiennes de courbes. , 1996 .

[25]  Gerhard Frey,et al.  Curves of genus 2 covering elliptic curves and an arithmetical application , 1991 .

[26]  F. Leprévost.  Torsion sur des familles de courbes de genre g , 1992 .

[27]  N. Katz,et al.  Arithmetic Moduli of Elliptic Curves. (AM-108) , 1985 .

[28]  Jean-François Mestre,et al.  Formules explicites et minoration de conducteurs de vari'et'es alg'ebriques , 1986 .

[29]  J. Cremona.  Algorithms for Modular Elliptic Curves , 1992 .

[30]  Edward F. Schaefer.  2-Descent on the Jacobians of Hyperelliptic Curves , 1995 .

[31]  Barry Mazur,et al.  Modular curves and the eisenstein ideal , 1977 .

[32]  Armand Brumer,et al.  The rank of elliptic curves , 1977 .

[33]  Bin Zhang.  Sur les jacobiennes des courbes à singularités ordinaires , 1997 .

[34]  F. Leprévost.  Sur certains sous-groupes de torsion de jacobiennes de courbes hyperelliptiques de genreg ≥ 1 , 1997 .

[35]  F. Leprévost.  Familles de courbes de genre 2 munies d'une classe de diviseurs rationnels d'ordre 15, 17, 19 ou 21 , 1991 .

[36]  D. Kubert.  Universal Bounds on the Torsion of Elliptic Curves , 1976 .

[37]  F. Momose,et al.  Torsion points on elliptic curves defined over quadratic fields , 1975, Nagoya Mathematical Journal.