Torsion and Tamagawa numbers
[Torsion et nombres de Tamagawa]
Annales de l'Institut Fourier, Tome 61 (2011) no. 5, pp. 1995-2037.

Soit K un corps de nombres, et soit A/K une variété abélienne. Dénotons par c le produit des nombres de Tamagawa de A/K, et par A(K) tors le sous-groupe fini des éléments de torsion de A(K). Le quotient c/|A(K) tors | apparaît dans la conjecture de Birch et Swinnerton-Dyer comme un facteur de la valeur du premier terme non-nul dans le développement limité en s=1 de la fonction L de A/K. Nous nous intéressons dans cet article aux diviseurs communs des entiers c et |A(K) tors |. Nous obtenons des résultats précis pour les courbes elliptiques sur ou sur une extension quadratique, et pour les surfaces abéliennes sur . La plus petite valeur de la fraction c/|E() tors | pour les courbes elliptiques sur est 1/5, obtenue seulement par la courbe modulaire X 1 (11)/.

Let K be a number field, and let A/K be an abelian variety. Let c denote the product of the Tamagawa numbers of A/K, and let A(K) tors denote the finite torsion subgroup of A(K). The quotient c/|A(K) tors | is a factor appearing in the leading term of the L-function of A/K in the conjecture of Birch and Swinnerton-Dyer. We investigate in this article possible cancellations in this ratio. Precise results are obtained for elliptic curves over or quadratic extensions K/, and for abelian surfaces A/. The smallest possible ratio c/|E() tors | for elliptic curves over is 1/5, achieved only by the modular curve X 1 (11).

DOI : 10.5802/aif.2664
Classification : 11G05, 11G10, 11G30, 11G35, 11G40, 14G05, 14G10
Mots clés : Abelian variety over a global field, torsion subgroup, Tamagawa number, elliptic curve, abelian surface, dual abelian variety, Weil restriction
Lorenzini, Dino 1

1 University of Georgia Department of mathematics Athens, GA 30602 (USA)
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Lorenzini, Dino. Torsion and Tamagawa numbers. Annales de l'Institut Fourier, Tome 61 (2011) no. 5, pp. 1995-2037. doi : 10.5802/aif.2664. http://archive.numdam.org/articles/10.5802/aif.2664/

[1] Agashe, Amod; Stein, William Visible evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of analytic rank zero, Math. Comp., Volume 74 (2005) no. 249, pp. 455-484 (With an appendix by J. Cremona and B. Mazur) | DOI | MR | Zbl

[2] Beauville, Arnaud Les familles stables de courbes elliptiques sur P 1 admettant quatre fibres singulières, C. R. Acad. Sci. Paris Sér. I Math., Volume 294 (1982) no. 19, pp. 657-660 | MR | Zbl

[3] Bégueri, Lucile Dualité sur un corps local à corps résiduel algébriquement clos, Mém. Soc. Math. France (N.S.) (1980/81) no. 4, pp. 121 | Numdam | MR | Zbl

[4] Beukers, F.; Schlickewei, H. P. The equation x+y=1 in finitely generated groups, Acta Arith., Volume 78 (1996) no. 2, pp. 189-199 | MR | Zbl

[5] Bosch, Siegfried; Liu, Qing Rational points of the group of components of a Néron model, Manuscripta Math., Volume 98 (1999) no. 3, pp. 275-293 | DOI | MR | Zbl

[6] Bosch, Siegfried; Lorenzini, Dino Grothendieck’s pairing on component groups of Jacobians, Invent. Math., Volume 148 (2002) no. 2, pp. 353-396 | DOI | MR | Zbl

[7] Bosch, Siegfried; Lütkebohmert, Werner; Raynaud, Michel Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 21, Springer-Verlag, Berlin, 1990 | MR | Zbl

[8] Bosma, Wieb; Cannon, John; Playoust, Catherine The Magma algebra system. I. The user language, J. Symbolic Comput., Volume 24 (1997) no. 3-4, pp. 235-265 Computational algebra and number theory (London, 1993), http://magma.maths.usyd.edu.au/magma/ | DOI | MR | Zbl

[9] Brumer, A.; Kramer, K. Paramodular abelian varieties of odd conductor (2010) (arXiv:1004.4699)

[10] Clark, Pete L.; Xarles, Xavier Local bounds for torsion points on abelian varieties, Canad. J. Math., Volume 60 (2008) no. 3, pp. 532-555 | DOI | MR | Zbl

[11] Conrad, Brian; Edixhoven, Bas; Stein, William J 1 (p) has connected fibers, Doc. Math., Volume 8 (2003), p. 331-408 (electronic) | MR | Zbl

[12] Cremona, J. E. Algorithms for modular elliptic curves, Cambridge University Press, Cambridge, 1997 | MR | Zbl

[13] Diem, C. A Study on Theoretical and Practical Aspects of Weil-Restrictions of Varieties Dissertation (Essen), http://www.math.uni-leipzig.de/~diem/preprints/english.html | Zbl

[14] Dummigan, Neil Rational points of order 7, Bull. Lond. Math. Soc., Volume 40 (2008) no. 6, pp. 1091-1093 | DOI | MR | Zbl

[15] Edixhoven, Bas Néron models and tame ramification, Compositio Math., Volume 81 (1992) no. 3, pp. 291-306 | Numdam | MR | Zbl

[16] Edixhoven, Bas; Liu, Qing; Lorenzini, Dino The p-part of the group of components of a Néron model, J. Algebraic Geom., Volume 5 (1996) no. 4, pp. 801-813 | MR | Zbl

[17] Elkies, N. Curves of genus 2 over whose Jacobians are absolutely simple abelian surfaces with torsion points of high order (http://www.math.harvard.edu/~elkies/g2_tors.html)

[18] Elkies, N. Elliptic curves in nature (http://math.harvard.edu/~elkies/nature.html)

[19] Elkies, N. Examples of high-order torsion points on simple genus-2 Jacobians (Manuscript, April 2001)

[20] Emerton, Matthew Optimal quotients of modular Jacobians, Math. Ann., Volume 327 (2003) no. 3, pp. 429-458 | DOI | MR | Zbl

[21] Erdös, P. Arithmetical properties of polynomials, J. London Math. Soc., Volume 28 (1953), pp. 416-425 | DOI | MR | Zbl

[22] Evertse, J.-H.; Schlickewei, H. P.; Schmidt, W. M. Linear equations in variables which lie in a multiplicative group, Ann. of Math. (2), Volume 155 (2002) no. 3, pp. 807-836 | DOI | MR | Zbl

[23] Flynn, E. V. Large rational torsion on abelian varieties, J. Number Theory, Volume 36 (1990) no. 3, pp. 257-265 | DOI | MR | Zbl

[24] Flynn, E. V. Large rational torsion on abelian varieties, J. Number Theory, Volume 36 (1990) no. 3, pp. 257-265 | DOI | MR | Zbl

[25] Fontaine, Jean-Marc Il n’y a pas de variété abélienne sur Z, Invent. Math., Volume 81 (1985) no. 3, pp. 515-538 | DOI | MR | Zbl

[26] Gonzalez-Aviles, C. On Néron class group of abelian varieties (Preprint 2009, arXiv:0909.4803v2 [math.NT] 5 Oct 2009)

[27] González-Jiménez, Enrique; González, Josep Modular curves of genus 2, Math. Comp., Volume 72 (2003) no. 241, p. 397-418 (electronic) | DOI | MR | Zbl

[28] Grothendieck, A. Éléments de géométrie algébrique. Étude locale des schémas et des morphismes de schémas, Inst. Hautes Études Sci. Publ. Math. (1966-1967) no. 24, 28, 32, pp. 231, 255, 361 | Numdam | Zbl

[29] Grothendieck, A. Groupes de monodromie en géométrie algébrique. I, Lecture Notes in Mathematics, Vol. 288, Springer-Verlag, Berlin, 1972 Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 I), Dirigé par A. Grothendieck. Avec la collaboration de M. Raynaud et D. S. Rim | MR

[30] Hindry, Marc; Silverman, Joseph H. Diophantine geometry, Graduate Texts in Mathematics, 201, Springer-Verlag, New York, 2000 (An introduction) | MR | Zbl

[31] Howe, Everett W.; Leprévost, Franck; Poonen, Bjorn Large torsion subgroups of split Jacobians of curves of genus two or three, Forum Math., Volume 12 (2000) no. 3, pp. 315-364 | DOI | MR | Zbl

[32] Kamienny, S. Torsion points on elliptic curves and q-coefficients of modular forms, Invent. Math., Volume 109 (1992) no. 2, pp. 221-229 | DOI | MR | Zbl

[33] Kenku, M. A.; Momose, F. Torsion points on elliptic curves defined over quadratic fields, Nagoya Math. J., Volume 109 (1988), pp. 125-149 | MR | Zbl

[34] Krumm, D. Tamagawa numbers of elliptic curves over cubic fields (in preparation)

[35] Kubert, Daniel Sion Universal bounds on the torsion of elliptic curves, Proc. London Math. Soc. (3), Volume 33 (1976) no. 2, pp. 193-237 | DOI | MR | Zbl

[36] Kunyavskiĭ, Boris; Sansuc, Jean-Jacques Réduction des groupes algébriques commutatifs, J. Math. Soc. Japan, Volume 53 (2001) no. 2, pp. 457-483 | DOI | MR | Zbl

[37] Leprévost, Franck Torsion sur des familles de courbes de genre g, Manuscripta Math., Volume 75 (1992) no. 3, pp. 303-326 | DOI | MR | Zbl

[38] Leprévost, Franck Jacobiennes de certaines courbes de genre 2: torsion et simplicité, J. Théor. Nombres Bordeaux, Volume 7 (1995) no. 1, pp. 283-306 Les Dix-huitièmes Journées Arithmétiques (Bordeaux, 1993) | DOI | Numdam | MR | Zbl

[39] Leprévost, Franck Sur certains sous-groupes de torsion de jacobiennes de courbes hyperelliptiques de genre g1, Manuscripta Math., Volume 92 (1997) no. 1, pp. 47-63 | DOI | MR | Zbl

[40] Ling, San On the Q-rational cuspidal subgroup and the component group of J 0 (p r ), Israel J. Math., Volume 99 (1997), pp. 29-54 | DOI | MR | Zbl

[41] Liu, Qing Courbes stables de genre 2 et leur schéma de modules, Math. Ann., Volume 295 (1993) no. 2, pp. 201-222 | DOI | MR | Zbl

[42] Liu, Qing Modèles minimaux des courbes de genre deux, J. Reine Angew. Math., Volume 453 (1994), pp. 137-164 | DOI | MR | Zbl

[43] Lorenzini, Dino J. On the group of components of a Néron model, J. Reine Angew. Math., Volume 445 (1993), pp. 109-160 | DOI | MR | Zbl

[44] Lorenzini, Dino J. Torsion points on the modular Jacobian J 0 (N), Compositio Math., Volume 96 (1995) no. 2, pp. 149-172 | Numdam | MR | Zbl

[45] Lorenzini, Dino J. Models of curves and wild ramification, Pure Appl. Math. Q., Volume 6 (2010) no. 1, Special Issue: In honor of John Tate. Part 2, pp. 41-82 | MR | Zbl

[46] Mazur, Barry Rational points of abelian varieties with values in towers of number fields, Invent. Math., Volume 18 (1972), pp. 183-266 | DOI | MR | Zbl

[47] Mazur, Barry Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. (1977) no. 47, p. 33-186 (1978) | DOI | Numdam | MR | Zbl

[48] McCallum, William G. Duality theorems for Néron models, Duke Math. J., Volume 53 (1986) no. 4, pp. 1093-1124 | DOI | MR | Zbl

[49] McCallum, William G. On the method of Coleman and Chabauty, Math. Ann., Volume 299 (1994) no. 3, pp. 565-596 | DOI | MR | Zbl

[50] Mihăilescu, Preda Primary cyclotomic units and a proof of Catalan’s conjecture, J. Reine Angew. Math., Volume 572 (2004), pp. 167-195 | DOI | MR | Zbl

[51] Milne, J. S. On the arithmetic of abelian varieties, Invent. Math., Volume 17 (1972), pp. 177-190 | DOI | MR | Zbl

[52] Milne, J. S. Arithmetic duality theorems, BookSurge, LLC, Charleston, SC, 2006 | MR | Zbl

[53] Müller, Hans H.; Ströher, Harald; Zimmer, Horst G. Torsion groups of elliptic curves with integral j-invariant over quadratic fields, J. Reine Angew. Math., Volume 397 (1989), pp. 100-161 | MR | Zbl

[54] Murabayashi, Naoki On normal forms of modular curves of genus 2, Osaka J. Math., Volume 29 (1992) no. 2, pp. 405-418 | MR | Zbl

[55] Nagell, T. Les points exceptionnels rationnels sur certaines cubiques du premier genre, Acta Arith., Volume 5 (1959), pp. 333-357 | MR | Zbl

[56] Namikawa, Yukihiko; Ueno, Kenji On fibres in families of curves of genus two. I. Singular fibres of elliptic type, Number theory, algebraic geometry and commutative algebra, in honor of Yasuo Akizuki, Kinokuniya, Tokyo, 1973, pp. 297-371 | MR | Zbl

[57] Oesterlé, Joseph Nombres de Tamagawa et groupes unipotents en caractéristique p, Invent. Math., Volume 78 (1984) no. 1, pp. 13-88 | DOI | MR | Zbl

[58] Ogawa, Hiroyuki Curves of genus 2 with a rational torsion divisor of order 23, Proc. Japan Acad. Ser. A Math. Sci., Volume 70 (1994) no. 9, pp. 295-298 | DOI | MR | Zbl

[59] Oort, Frans Subvarieties of moduli spaces, Invent. Math., Volume 24 (1974), pp. 95-119 | DOI | MR | Zbl

[60] Patterson, Roger D.; van der Poorten, Alfred J.; Williams, Hugh C. Sequences of Jacobian varieties with torsion divisors of quadratic order, Funct. Approx. Comment. Math., Volume 39 (2008) no. part 2, pp. 345-360 | DOI | MR | Zbl

[61] Penniston, David Unipotent groups and curves of genus two, Math. Ann., Volume 317 (2000) no. 1, pp. 57-78 | DOI | MR | Zbl

[62] Poor, C.; Yuen, D. Paramodular cusp forms Preprint (2009), available at http://arxiv.org/PS_cache/arxiv/pdf/0912/0912.0049v1.pdf

[63] Reichert, Markus A. Explicit determination of nontrivial torsion structures of elliptic curves over quadratic number fields, Math. Comp., Volume 46 (1986) no. 174, pp. 637-658 | DOI | MR | Zbl

[64] Sage Mathematics Software (http://www.sagemath.org/)

[65] Silverman, Joseph H. Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, 151, Springer-Verlag, New York, 1994 | MR | Zbl

[66] Stevens, Glenn Stickelberger elements and modular parametrizations of elliptic curves, Invent. Math., Volume 98 (1989) no. 1, pp. 75-106 | DOI | MR | Zbl

[67] Tate, J. Algorithm for determining the type of a singular fiber in an elliptic pencil, Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Springer, Berlin, 1975, p. 33-52. Lecture Notes in Math., Vol. 476 | MR | Zbl

[68] Tzermias, Pavlos Torsion parts of Mordell-Weil groups of Fermat Jacobians, Internat. Math. Res. Notices (1998) no. 7, pp. 359-369 | DOI | MR | Zbl

[69] Vatsal, V. Multiplicative subgroups of J 0 (N) and applications to elliptic curves, J. Inst. Math. Jussieu, Volume 4 (2005) no. 2, pp. 281-316 | DOI | MR | Zbl

[70] Yang, Yifan Modular units and cuspidal divisor class groups of X 1 (N), J. Algebra, Volume 322 (2009) no. 2, pp. 514-553 | DOI | MR | Zbl

[71] Zarhin, Yuri G. Hyperelliptic Jacobians without complex multiplication, Math. Res. Lett., Volume 7 (2000) no. 1, pp. 123-132 | MR | Zbl

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