On the asymptotic behaviour of Heegner points

We prove that all but finitely many Heegner points on a given modular elliptic curve (or, more generally, on a given quotient of the modularJacobian variety $J_0$(N)) are of infinite order in the Mordell-Weil group where they naturally live, i.e., over the corresponding ring class field.

PDF

___

[1] M. Bertolini, Selmer groups and Heegner points in anticyclotomic Zp-extensions, Compositio Math. 99 (1995), 153–182
[2] M. Bertolini, Growth of Mordell-Weil groups in anticyclotomic towers; in: Arithmetic geometry (Cortona, 1994; F. Catanese, ed.), Sympos. Math., XXXVII, Cambridge Univ. Press, Cambridge, 1997, pp. 23–44
[3] M. Bertolini and H. Darmon, Non-triviality of families of Heegner points and ranks of Selmer groups over anticyclotomic towers. J. Ramanujan Math. Soc. 13 (1998), 15–24
[4] B.H. Gross and D. Zagier, Heegner points and the derivatives of L-series, Inventiones Math. 84 (1986), 225–320
[5] V.A. Kolyvagin, On the structure of Selmer groups, Mathematische Annalen 291 (1991), 253–259
[6] P.F. Kurcanov, Elliptic Curves of infinite rank over Γ-extensions, Math. USSR Sbornik 19 (1973), 320–324
[7] Yu. Manin, Krugovye polja i moduljarnye krivye, Uspechi Mat. Nauk, 26 (1971), 7–71. English translation: Cyclotomic fields and modular curves, Russian Math. Surveys 26, no. 6 (1971), 7–78
[8] B. Mazur, Rational points on abelian varieties with values in towers of number fields, Inventiones Math. 18 (1972), 183–266
[9] B. Mazur, Modular Curves and Arithmetic, Proc. ICM Warszawa 1983, vol. I, pp. 185–211
[10] B. Perrin-Riou, Fonctions L p-adiques, theorie d’Iwasawa et points de Heegner, Bull. Soc. Math. France 115 (1987), 399–456
[11] D. Rohrlich, Nonvanishing of L-functions for GL(2), Inventiones Math. 97 (1989), 381–403
[12] C. Schoen, Complex multiplication cycles on elliptic modular threefolds, Duke Math. J. 53 (1986), 771–794

ISSN: 1300-0098
Yayın Aralığı: Yılda 6 Sayı
Yayıncı: TÜBİTAK

Arşiv