Skip to main content Accessibility help
×
Home
Hostname: page-component-78bd46657c-2pqp7 Total loading time: 0.457 Render date: 2021-05-08T14:23:01.091Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

Sur le rang des courbes elliptiques sur les corps de classes de Hilbert

Published online by Cambridge University Press:  10 February 2011

Nicolas Templier
Affiliation:
Institute for Advanced Study, Princeton, NJ 08540, USA (email: nicolas.templier@normalesup.org)
Rights & Permissions[Opens in a new window]

Abstract

Let E/ℚ be an elliptic curve and let D<0 be a sufficiently large fundamental discriminant. If contains Heegner points of discriminant D, those points generate a subgroup of rank at least |D|δ, where δ>0 is an absolute constant. This result is compatible with the Birch and Swinnerton-Dyer conjecture.

Résumé

Soit E/ℚ une courbe elliptique. Soit D<0 un discriminant fondamental suffisamment grand. Si contient des points de Heegner de discriminant D, ces points engendrent un sous-groupe dont le rang est supérieur à |D|δ, où δ>0 est une constante absolue. Ce résultat est en accord avec la conjecture de Birch et Swinnerton-Dyer.

Type
Research Article
Copyright
Copyright © Foundation Compositio Mathematica 2011

References

[AN10]Aflalo, E. and Nekovář, J., Non-triviality of CM points in ring class field towers, Israel J. Math. 175 (2010), 225284.CrossRefGoogle Scholar
[BCHIS66]Borel, A., Chowla, S., Herz, C. S., Iwasawa, K. and Serre, J.-P., Seminar on complex multiplication, in Seminar held at the Institute for Advanced Study, Princeton, NJ, 1957–1958, Lecture Notes in Mathematics, vol. 21 (Springer, Berlin, 1966).Google Scholar
[BCDT01]Breuil, C., Conrad, B., Diamond, F. and Taylor, R., On the modularity of elliptic curves over Q: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), 843939.CrossRefGoogle Scholar
[BP09]Buium, A. and Poonen, B., Independence of points on elliptic curves arising from special points on modular and Shimura curves. I. Global results, Duke Math. J. 147 (2009), 181191.CrossRefGoogle Scholar
[BFH90]Bump, D., Friedberg, S. and Hoffstein, J., Nonvanishing theorems for L-functions of modular forms and their derivatives, Invent. Math. 102 (1990), 543618.CrossRefGoogle Scholar
[Cor02a]Cornut, C., Mazur’s conjecture on higher Heegner points, Invent. Math. 148 (2002), 495523.CrossRefGoogle Scholar
[Cor02b]Cornut, C., Non-trivialité des points de Heegner, C. R. Math. Acad. Sci. Paris 334 (2002), 10391042.CrossRefGoogle Scholar
[CV05]Cornut, C. and Vatsal, V., CM points and quaternion algebras, Doc. Math. 10 (2005), 263309 (electronic).Google Scholar
[CV07]Cornut, C. and Vatsal, V., Nontriviality of Rankin–Selberg L-functions and CM points, in L-functions and Galois representations, London Mathematical Society Lecture Note Series, vol. 320 (Cambridge University Press, Cambridge, 2007), 121186.CrossRefGoogle Scholar
[Dar01]Darmon, H., Integration on ℋp×ℋ and arithmetic applications, Ann. of Math. (2) 154 (2001), 589639.CrossRefGoogle Scholar
[Dar04]Darmon, H., Rational points on modular elliptic curves, CBMS Regional Conference Series in Mathematics, vol. 101 (Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2004).Google Scholar
[Dar06]Darmon, H., Heegner points, Stark–Heegner points, and values of L-series, in International congress of mathematicians, Vol. II (Eur. Math. Soc. Zürich, 2006), 313345.Google Scholar
[Duk88]Duke, W., Hyperbolic distribution problems and half-integral weight Maass forms, Invent. Math. 92 (1988), 7390.CrossRefGoogle Scholar
[DFI95]Duke, W., Friedlander, J. and Iwaniec, H., Class group L-functions, Duke Math. J. 79 (1995), 156.CrossRefGoogle Scholar
[Gro88]Gross, B., Local orders, root numbers, and modular curves, Amer. J. Math. 110 (1988), 11531182.CrossRefGoogle Scholar
[GP91]Gross, B. and Prasad, D., Test vectors for linear forms, Math. Ann. 291 (1991), 343355.CrossRefGoogle Scholar
[GZ86]Gross, B. and Zagier, D., Heegner points and derivatives of L-series, Invent. Math. 84 (1986), 225320.CrossRefGoogle Scholar
[HPS89]Hijikata, H., Pizer, A. and Shemanske, T., Orders in quaternion algebras, J. Reine Angew. Math. 394 (1989), 59106.Google Scholar
[Iwa90]Iwaniec, H., On the order of vanishing of modular L-functions at the critical point, Sém. Théor. Nombres Bordeaux (2) 2 (1990), 365376.CrossRefGoogle Scholar
[Jac72]Jacquet, H., Automorphic forms on GL(2). Part II, Lecture Notes in Mathematics, vol. 278 (Springer, Berlin, 1972).CrossRefGoogle Scholar
[JL70]Jacquet, H. and Langlands, R., Automorphic forms on GL(2) (Springer, Berlin, 1970).CrossRefGoogle Scholar
[KS99]Katz, N. and Sarnak, P., Random matrices, Frobenius eigenvalues, and monodromy, American Mathematical Society Colloquium Publications, vol. 45 (American Mathematical Society, Providence, RI, 1999).Google Scholar
[Kol88a]Kolyvagin, V., Finiteness of E(Q) and SH(E,Q) for a subclass of Weil curves, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), 522540, 670–671.Google Scholar
[Kol88b]Kolyvagin, V., The Mordell–Weil and Shafarevich–Tate groups for Weil elliptic curves, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), 11541180, 1327.Google Scholar
[Maz84]Mazur, B., Modular curves and arithmetic, in Proceedings of the International Congress of Mathematicians (Warsaw 1983), Vol. 1, 2 (PWN, Warsaw, 1984), 185211.Google Scholar
[MS74]Mazur, B. and Swinnerton-Dyer, P., Arithmetic of Weil curves, Invent. Math. 25 (1974), 161.CrossRefGoogle Scholar
[Mic04]Michel, Ph., The subconvexity problem for Rankin–Selberg L-functions and equidistribution of Heegner points, Ann. of Math. (2) 160 (2004), 185236.CrossRefGoogle Scholar
[MV06]Michel, Ph. and Venkatesh, A., Equidistribution, L-functions and ergodic theory: on some problems of Yu. Linnik, in International congress of mathematicians, Vol. II (Eur. Math. Soc. Zürich, 2006), 421457.Google Scholar
[MV07]Michel, Ph. and Venkatesh, A., Heegner points and non-vanishing of Rankin/SelbergL-functions, in Analytic number theory, Clay Mathematics Proceedings, vol. 7 (American Mathematical Society, Providence, RI, 2007), 169183.Google Scholar
[MR82]Montgomery, H. and Rohrlich, D., On the L-functions of canonical Hecke characters of imaginary quadratic fields. II, Duke Math. J. 49 (1982), 937942.CrossRefGoogle Scholar
[MM91]Murty, M. and Murty, V., Mean values of derivatives of modular L-series, Ann. of Math. (2) 133 (1991), 447475.CrossRefGoogle Scholar
[Nek07]Nekovář, J., The Euler system method for CM points on Shimura curves, in L-functions and Galois representations, London Mathematical Society Lecture Note Series, vol. 320 (Cambridge University Press, Cambridge, 2007), 471547.CrossRefGoogle Scholar
[NS99]Nekovář, J. and Schappacher, N., On the asymptotic behaviour of Heegner points, Turkish J. Math. 23 (1999), 549556.Google Scholar
[PR94]Platonov, V. and Rapinchuk, A., Algebraic groups and number theory, Pure and Applied Mathematics, vol. 139 (Academic Press, Boston, MA, 1994), Translated from the 1991 Russian original by Rachel Rowen.Google Scholar
[Pra07]Prasad, D., Relating invariant linear form and local epsilon factors via global methods, Duke Math. J. 138 (2007), 233261.CrossRefGoogle Scholar
[Rib92]Ribet, K., Abelian varieties over Q and modular forms, in Algebra and topology 1992 (Taejŏn) (Korea Advanced Institute of Science and Technology, Taejŏn, 1992), 5379.Google Scholar
[RT09]Ricotta, G. and Templier, N., Comportement asymptotique des hauteurs des points de Heegner, J. Théor. Nombres Bordeaux 21 (2009), 741753.CrossRefGoogle Scholar
[RV08]Ricotta, G. and Vidick, T., Hauteur asymptotique des points de Heegner, Canad. J. Math. 60 (2008), 14061436 (in French, with English summary).CrossRefGoogle Scholar
[Roh80a]Rohrlich, D., Galois conjugacy of unramified twists of Hecke characters, Duke Math. J. 47 (1980), 695703.CrossRefGoogle Scholar
[Roh80b]Rohrlich, D., The nonvanishing of certain Hecke L-functions at the center of the critical strip, Duke Math. J. 47 (1980), 223232.CrossRefGoogle Scholar
[Roh80c]Rohrlich, D., On the L-functions of canonical Hecke characters of imaginary quadratic fields, Duke Math. J. 47 (1980), 547557.CrossRefGoogle Scholar
[RS07]Rosen, M. and Silverman, J., On the independence of Heegner points associated to distinct quadratic imaginary fields, J. Number Theory 127 (2007), 1036.CrossRefGoogle Scholar
[Sie35]Siegel, C.-L., Über die Classenzahl quadratischer Zahlkörper, Acta Arith. 1 (1935), 8386 (reprinted in Ges. Abh. I, 406–409, Springer, Berlin, 1966).CrossRefGoogle Scholar
[Sil09]Silverman, J., The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, second edition (Springer, New York, 2009), corrected reprint of the 1986 original.CrossRefGoogle Scholar
[SUZ97]Szpiro, L., Ullmo, E. and Zhang, S., Équirépartition des petits points, Invent. Math. 127 (1997), 337347.CrossRefGoogle Scholar
[TW95]Taylor, R. and Wiles, A., Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), 553572.CrossRefGoogle Scholar
[Tem07]Templier, N., Heegner points and Eisenstein series, Preprint (2007), available at http://arxiv.org/abs/0808.1476, Forum Math., to appear.Google Scholar
[Tem08a]Templier, N., Minoration de rangs de courbes elliptiques, C. R. Math. Acad. Sci. Paris 346 (2008), 12251230 (in French, with English and French summaries).CrossRefGoogle Scholar
[Tem08b]Templier, N., A non-split sum of coefficients of modular forms, Preprint (2008), available at arXiv:0902.2496, Duke Math. J., to appear.Google Scholar
[Tun83]Tunnell, J., Local ϵ-factors and characters of GL(2), Amer. J. Math. 105 (1983), 12771307.CrossRefGoogle Scholar
[Ull98]Ullmo, E., Positivité et discrétion des points algébriques des courbes, Ann. of Math. (2) 147 (1998), 167179.CrossRefGoogle Scholar
[Vat03]Vatsal, V., Special values of anticyclotomic L-functions, Duke Math. J. 116 (2003), 219261.CrossRefGoogle Scholar
[Wal85]Waldspurger, J.-L., Sur les valeurs de certaines fonctions L automorphes en leur centre de symétrie, Compositio Math. 54 (1985), 173242.Google Scholar
[Wil95]Wiles, A., Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), 443551.CrossRefGoogle Scholar
[Win07]Wintenberger, J.-P., La conjecture de modularité de Serre : le cas de conducteur (d’après C. Khare), Astérisque (2007), Exp. No. 956, viii, 99–121, Sém. Bourbaki. Vol. 2005–2006.Google Scholar
[YZZ]Yuan, X., Zhang, S.-W. and Zhang, W., Gross–Zagier formula, Ann. of Math. Stud., to appear.Google Scholar
[Zha98]Zhang, S.-W., Equidistribution of small points on abelian varieties, Ann. of Math. (2) 147 (1998), 159165.CrossRefGoogle Scholar
[Zha01a]Zhang, S.-W., Heights of Heegner points on Shimura curves, Ann. of Math. (2) 153 (2001), 27147.CrossRefGoogle Scholar
[Zha01b]Zhang, S.-W., Gross–Zagier formula for GL 2, Asian J. Math. 5 (2001), 183290.CrossRefGoogle Scholar
[Zha05]Zhang, S.-W., Equidistribution of CM-points on quaternion Shimura varieties, Int. Math. Res. Not. (2005), 36573689.CrossRefGoogle Scholar
You have Access

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Sur le rang des courbes elliptiques sur les corps de classes de Hilbert
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Sur le rang des courbes elliptiques sur les corps de classes de Hilbert
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Sur le rang des courbes elliptiques sur les corps de classes de Hilbert
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *