Geodesic
Heegner Points on Cartan Non-split Curves
Canadian journal of mathematics, Tome 68 (2016) no. 2, pp. 422-444

Voir la notice de l'article provenant de la source Cambridge University Press

Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$ , and let $K$ be an imaginary quadratic field such that the root number of $E/K$ is −1. Let $O$ be an order in $K$ and assume that there exists an odd prime $p$ such that ${{p}^{2}}\,\parallel \,N$ , and $p$ is inert in $O$ . Although there are no Heegner points on ${{X}_{0}}(N)$ attached to $O$ , in this article we construct such points on Cartan non-split curves. In order to do that, we give a method to compute Fourier expansions for forms on Cartan non-split curves, and prove that the constructed points form a Heegner system as in the classical case.
DOI : 10.4153/CJM-2015-047-6
Mots-clés : 11G05, 11F30, Cartan curves, Heegner points 
Kohen, Daniel; Pacetti, Ariel. Heegner Points on Cartan Non-split Curves. Canadian journal of mathematics, Tome 68 (2016) no. 2, pp. 422-444. doi: 10.4153/CJM-2015-047-6
@article{10_4153_CJM_2015_047_6,
     author = {Kohen, Daniel and Pacetti, Ariel},
     title = {Heegner {Points} on {Cartan} {Non-split} {Curves}},
     journal = {Canadian journal of mathematics},
     pages = {422--444},
     year = {2016},
     volume = {68},
     number = {2},
     doi = {10.4153/CJM-2015-047-6},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2015-047-6/}
}
TY  - JOUR
AU  - Kohen, Daniel
AU  - Pacetti, Ariel
TI  - Heegner Points on Cartan Non-split Curves
JO  - Canadian journal of mathematics
PY  - 2016
SP  - 422
EP  - 444
VL  - 68
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-2015-047-6/
DO  - 10.4153/CJM-2015-047-6
ID  - 10_4153_CJM_2015_047_6
ER  - 
%0 Journal Article
%A Kohen, Daniel
%A Pacetti, Ariel
%T Heegner Points on Cartan Non-split Curves
%J Canadian journal of mathematics
%D 2016
%P 422-444
%V 68
%N 2
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-2015-047-6/
%R 10.4153/CJM-2015-047-6
%F 10_4153_CJM_2015_047_6

[AL78] [AL78] Atkin, A. O. L. and Li, W. C. W., Twists of newforms and pseudo-eigenvalues of W -operators. Invent. Math. 48(1978), no. 3, 221–243. http://dx.doi.Org/10.1007/BF01390245http://dx.doi.org/10.1007/BF01390245 Google Scholar

[Che98] [Che98] Chen, I., The Jacobians of non-split Cartan modular curves. Proc. London Math. Soc. 77(1998), 1–38. Google Scholar | DOI

[Dar04] [Dar04] Darmon, H., Rational points on modular elliptic curves. CBMS Regional Conference Series in Mathematics, 101, American Mathematical Society, Providence, RI, 2004. Google Scholar

[DD11] [DD11] Dokchitser, T. and Dokchitser, V., Euler factors determine local Weil representations. arxiv:1112.4889 Google Scholar

[DS05] [DS05] Diamond, F. and Shurman, J., A first course in modular forms. Graduate Texts in Mathematics, 228, Springer-Verlag, New York, 2005. Google Scholar

[dSE00] [dSE00] de Smit, B. and Edixhoven, B., Sur un résultat d'Imin Chen. Math. Res. Lett. 7(2000), no. 2–3, 147–153. Google Scholar | DOI

[Edi96] [Edi96] Edixhoven, B., On a result oflmin Chen. arxiv:alg-geom/9604008 Google Scholar

[Gro84] [Gro84] Gross, B. H., Heegner points on X0(N). In: Modular forms (Durham, 1983), Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., Horwood, Chichester, 1984, pp. 87–105. Google Scholar

[Gro91] [Gro91] Gross, B. H., Kolyvagin's work on modular elliptic curves. In: L-functions and arithmetic (Durham, 1989), London Math. Soc. Lecture Note Ser., 153, Cambridge University Press, Cambridge, 1991, pp. 235–256. Google Scholar | DOI

[GZ86] [GZ86] Gross, B. H. and Zagier, D. B., Heegner points and derivatives of L-series. Invent. Math. 84(1986), no. 2, 225–320. Google Scholar | DOI

[Lan87] [Lan87] Lang, S., Elliptic functions. Second ed., Graduate Texts in Mathematics, 112, Springer-Verlag, New York, 1987. Google Scholar | DOI

[Maz78] [Maz78] Mazur, B., Rational isogenies of prime degree (with an appendix by D. Goldfeld). Invent. Math. 44(1978), no. 2, 129–162. Google Scholar | DOI

[Pacl3] [Pacl3] Pacetti, A., On the change of root numbers under twisting and applications. Proc. Amer. Math. Soc. 141(2013), no. 8, 2615–2628. http://dx.doi.Org/10.1090/S0002-9939-2013-11532-7 Google Scholar

[PAR14] [PAR14] PARI Group, Bordeaux. PARI/GP version 2.7.0, 2014. http://pari.math.u-bordeaux.fr/ Google Scholar

[Raj98] [Raj98] Rajan, C. S., On strong multiplicity one for ℓ-adic representations. Internat. Math. Res. Notices 1998, no. 3, 161–172. Google Scholar | DOI

[RW14] [RW14] Rebolledo, M. and Wuthrich, C., A moduli interpretation for the non-split Cartan modular curve. http://arxiv:1402.3498 Google Scholar

[Ser67] [Ser67] Serre, J.-P., Complex multiplication. In: Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 292–296. Google Scholar

[Ser97] [Ser97] Serre, J.-P., Lectures on the Mordell-Weil theorem. Third ed., Aspects of Mathematics. Friedr. Vieweg & Sohn, Braunschweig, 1997. Google Scholar

[Shi94] [Shi94] Shimura, G., Introduction to the arithmetic theory of automorphic functions. Publications of the Mathematical Society of Japan, 11, Princeton University Press, Princeton, NJ, 1994. Google Scholar

[Zha01] [Zha01] Zhang, S.-W., Gross-Zagier formula for GL2. Asian J. Math. 5(2001), no. 2, 183–290. Google Scholar

[Zha04] [Zha04] Zhang, S.-W., Gross-Zagier formula for GL(2). II. In: Heegner points and Rankin L-series, Math. Sci. Res. Inst. Publ., 49, Cambridge University Press, Cambridge, 2004, pp. 191–214. Google Scholar | DOI

Cité par Sources :