Examples of torsion points on genus two curves

Boxall, John; Grant, David

doi:10.1090/S0002-9947-00-02368-0

Examples of torsion points on genus two curves
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by John Boxall and David Grant PDF

Trans. Amer. Math. Soc. 352 (2000), 4533-4555 Request permission

Abstract:

We describe a method that sometimes determines all the torsion points lying on a curve of genus two defined over a number field and embedded in its Jacobian using a Weierstrass point as base point. We then apply this to the examples $y^{2}=x^{5}+x$, $y^{2}=x^{5}+5 x^{3}+x$, and $y^{2}-y=x^{5}$.

References

Laurent Moret-Bailly, Hauteurs et classes de Chern sur les surfaces arithmétiques, Astérisque 183 (1990), 37–58 (French). Séminaire sur les Pinceaux de Courbes Elliptiques (Paris, 1988). MR 1065154
Alexandru Buium, Geometry of $p$-jets, Duke Math. J. 82 (1996), no. 2, 349–367. MR 1387233, DOI 10.1215/S0012-7094-96-08216-2
Hidegorô Nakano, Über Abelsche Ringe von Projektionsoperatoren, Proc. Phys.-Math. Soc. Japan (3) 21 (1939), 357–375 (German). MR 94
J. W. S. Cassels and E. V. Flynn, Prolegomena to a middlebrow arithmetic of curves of genus $2$, London Mathematical Society Lecture Note Series, vol. 230, Cambridge University Press, Cambridge, 1996. MR 1406090, DOI 10.1017/CBO9780511526084
Robert F. Coleman, Torsion points on curves and $p$-adic abelian integrals, Ann. of Math. (2) 121 (1985), no. 1, 111–168. MR 782557, DOI 10.2307/1971194
Robert F. Coleman, Torsion points on Fermat curves, Compositio Math. 58 (1986), no. 2, 191–208. MR 844409
Robert F. Coleman, Ramified torsion points on curves, Duke Math. J. 54 (1987), no. 2, 615–640. MR 899407, DOI 10.1215/S0012-7094-87-05425-1
R. F. Coleman, B. Kaskel, K. A. Ribet, Torsion points on $X_{0}(N)$, in Automorphic Forms, Automorphic Representations, and Arithmetic, Proc. Sympos. Pure Math., vol. 66, part 1, 1999, pp. 27–49.
Robert F. Coleman, Akio Tamagawa, and Pavlos Tzermias, The cuspidal torsion packet on the Fermat curve, J. Reine Angew. Math. 496 (1998), 73–81. MR 1605810, DOI 10.1515/crll.1998.033
J. E. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, Cambridge, 1992. MR 1201151
Sinnou David and Patrice Philippon, Minorations des hauteurs normalisées des sous-variétés de variétés abéliennes, Number theory (Tiruchirapalli, 1996) Contemp. Math., vol. 210, Amer. Math. Soc., Providence, RI, 1998, pp. 333–364 (French, with English and French summaries). MR 1478502, DOI 10.1090/conm/210/02795
Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
Gerhard Frey and Ernst Kani, Curves of genus $2$ covering elliptic curves and an arithmetical application, Arithmetic algebraic geometry (Texel, 1989) Progr. Math., vol. 89, Birkhäuser Boston, Boston, MA, 1991, pp. 153–176. MR 1085258, DOI 10.1007/978-1-4612-0457-2_{7}
David Grant, A proof of quintic reciprocity using the arithmetic of $y^2=x^5+1/4$, Acta Arith. 75 (1996), no. 4, 321–327. MR 1387868, DOI 10.4064/aa-75-4-321-337
D. Grant, Units from $5$-torsion on the Jacobian of $y^{2}=x^{5}+1/4$ and the conjectures of Stark and Rubin, J. Number Theory 77 (1999), 227–251.
David Grant, Units from $3$- and $4$-torsion on Jacobians of curves of genus $2$, Compositio Math. 94 (1994), no. 3, 311–320. MR 1310862
Ralph Greenberg, On the Jacobian variety of some algebraic curves, Compositio Math. 42 (1980/81), no. 3, 345–359. MR 607375
Marc Hindry, Autour d’une conjecture de Serge Lang, Invent. Math. 94 (1988), no. 3, 575–603 (French). MR 969244, DOI 10.1007/BF01394276
Jun-ichi Igusa, Arithmetic variety of moduli for genus two, Ann. of Math. (2) 72 (1960), 612–649. MR 114819, DOI 10.2307/1970233
Robert M. Kuhn, Curves of genus $2$ with split Jacobian, Trans. Amer. Math. Soc. 307 (1988), no. 1, 41–49. MR 936803, DOI 10.1090/S0002-9947-1988-0936803-3
Serge Lang, Division points on curves, Ann. Mat. Pura Appl. (4) 70 (1965), 229–234. MR 190146, DOI 10.1007/BF02410091
Anand Pillay, Model theory and Diophantine geometry, Bull. Amer. Math. Soc. (N.S.) 34 (1997), no. 4, 405–422. MR 1458425, DOI 10.1090/S0273-0979-97-00730-1
M. Raynaud, Courbes sur une variété abélienne et points de torsion, Invent. Math. 71 (1983), no. 1, 207–233 (French). MR 688265, DOI 10.1007/BF01393342
M. Raynaud, Sous-variétés d’une variété abélienne et points de torsion, Arithmetic and geometry, Vol. I, Progr. Math., vol. 35, Birkhäuser Boston, Boston, MA, 1983, pp. 327–352 (French). MR 717600
Jean-Pierre Serre, Abelian $l$-adic representations and elliptic curves, W. A. Benjamin, Inc., New York-Amsterdam, 1968. McGill University lecture notes written with the collaboration of Willem Kuyk and John Labute. MR 0263823
Jean-Pierre Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), no. 4, 259–331 (French). MR 387283, DOI 10.1007/BF01405086
D. Shaulis, Torsion points on the Jacobian of a hyperelliptic image of a Fermat curve, PhD. Thesis, University of Colorado at Boulder (1998).
Goro Shimura and Yutaka Taniyama, Complex multiplication of abelian varieties and its applications to number theory, Publications of the Mathematical Society of Japan, vol. 6, Mathematical Society of Japan, Tokyo, 1961. MR 0125113
Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210, DOI 10.1007/978-1-4757-1920-8
Rudolph E. Langer, The boundary problem of an ordinary linear differential system in the complex domain, Trans. Amer. Math. Soc. 46 (1939), 151–190 and Correction, 467 (1939). MR 84, DOI 10.1090/S0002-9947-1939-0000084-7
Emmanuel Ullmo, Positivité et discrétion des points algébriques des courbes, Ann. of Math. (2) 147 (1998), no. 1, 167–179 (French). MR 1609514, DOI 10.2307/120987
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1