The cuspidal torsion packet on hyperelliptic Fermat quotients
Journal de Théorie des Nombres de Bordeaux, Volume 16 (2004) no. 3, pp. 577-585.

Let 7 be a prime, C be the non-singular projective curve defined over by the affine model y(1-y)=x , the point of C at infinity on this model, J the Jacobian of C, and φ:CJ the albanese embedding with as base point. Let ¯ be an algebraic closure of . Taking care of a case not covered in [12], we show that φ(C)J tors ( ¯) consists only of the image under φ of the Weierstrass points of C and the points (x,y)=(0,0) and (0,1), where J tors denotes the torsion points of J.

Soit 7 un nombre premier, soit C la courbe projective lisse définie sur par le modèle affine y(1-y)=x , soit le point à l’infini de ce modèle de C, soit J la jacobienne de C et soit φ:CJ le morphisme d’Abel-Jacobi associé à . Soit ¯ une clôture algébrique de . Nous traitons ici un cas non couvert dans [12], en montrant que φ(C)J tors ( ¯) est composé de l’image par φ des points de Weierstrass de C ainsi que les points (x,y)=(0,0) et (0,1) de C. Ici, J tors désigne les points de torsion de J.

Published online:
DOI: 10.5802/jtnb.462
David Grant 1; Delphy Shaulis 1

1 Department of Mathematics University of Colorado at Boulder Boulder, CO 80309-0395 USA
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David Grant; Delphy Shaulis. The cuspidal torsion packet on hyperelliptic Fermat quotients. Journal de Théorie des Nombres de Bordeaux, Volume 16 (2004) no. 3, pp. 577-585. doi : 10.5802/jtnb.462. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.462/

[1] G. Anderson, Torsion points on Jacobians of quotients of Fermat curves and p-adic soliton theory. Invent. Math 118, (1994), 475–492. | MR | Zbl

[2] M. Baker, Torsion points on modular curves. Invent. Math 140, (2000), 487–509. | MR | Zbl

[3] M. Baker, B. Poonen, Torsion packets on curves. Compositio Math 127, (2001), 109–116. | MR | Zbl

[4] M. Baker, K. Ribet, Galois theory and torsion points on curves. Journal de Théorie des nombres de Bordeaux 15, (2003), 11–32. | Numdam | MR | Zbl

[5] J. Boxall, D. Grant, Examples of torsion points on genus 2 curves. Trans. Amer. Math. Soc 352, (2000), 4533–4555. | MR | Zbl

[6] J. Boxall, D. Grant, Singular torsion on elliptic curves. Mathematical Research Letters 10, (2003), 847–866. | MR | Zbl

[7] F. Calegari, Almost rational torsion points on semistable elliptic curves. IMRN no. 10, (2001), 487–503. | MR | Zbl

[8] J. Coates, A. Wiles, On the conjecture of Birch and Swinnerton-Dyer. Invent. Math 39, (1977), 223–251. | MR | Zbl

[9] R. F. Coleman, Torsion points on Fermat curves. Compositio Math 58, (1986), 191–208. | Numdam | MR | Zbl

[10] R. F. Coleman , Torsion points on abelian étale coverings of 1 -{0,1,}. Trans. AMS 311, (1989), 185–208. | MR | Zbl

[11] R. F. Coleman, W. McCallum, Stable reduction of Fermat curves and Jacobi sum Hecke characters J. Reine Angew. Math 385, (1988), 41–101. | EuDML | MR | Zbl

[12] R. F. Coleman, A. Tamagawa, P. Tzermias, The cuspidal torsion packet on the Fermat curve. J. Reine Angew. Math 496, (1998), 73–81. | MR | Zbl

[13] D. Grant, Torsion on theta divisors of hyperelliptic Fermat jacobians. Compositio Math. 140, (2004), 1432–1438. | MR | Zbl

[14] R. Greenberg, On the Jacobian variety of some algebraic curves. Compositio Math 42, (1981), 345–359. | EuDML | Numdam | MR | Zbl

[15] R. Gupta, Ramification in the Coates-Wiles tower. Invent. Math 81, (1985), 59–69. | EuDML | MR | Zbl

[16] M. Kurihara, Some remarks on conjectures about cyclotomic fields and K-groups of . Composition Math 81, (1992), 223–236. | EuDML | Numdam | MR | Zbl

[17] S. Lang, Division points on curves. Ann. Mat. Pura. Appl 70, (1965), 229-234. | MR | Zbl

[18] S. Lang, Complex Multiplication. Springer-Verlag, New York, 1983. | MR | Zbl

[19] K. Ribet, M. Kim, Torsion points on modular curves and Galois theory. Notes of a talk by K. Ribet in the Distinguished Lecture Series, Southwestern Center for Arithmetic Algebraic Geometry, (May 1999).

[20] D. Shaulis, Torsion points on the Jacobian of a hyperelliptic rational image of a Fermat curve. Thesis, University of Colorado at Boulder, 1998.

[21] B. Simon, Torsion points on a theta divisor in the Jacobian of a Fermat quotient. Thesis, University of Colorado at Boulder, 2003.

[22] A. Tamagawa, Ramification of torsion points on curves with ordinary semistable Jacobian varieties. Duke Math. J 106, (2001), 281–319. | MR | Zbl

[23] P. Tzermias, The Manin-Mumford conjecture: a brief survey. Bull. London Math. Soc. 32, (2000), 641–652. | MR | Zbl

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