Heegner cycles, modular forms and jacobi forms
Journal de Théorie des Nombres de Bordeaux, Tome 3 (1991) no. 1, pp. 93-116.

Nous présentons une interprétation géométrique d'une loi arithmétique pour déduire des formules explicites pour les coefficients des formes modulaires elliptiques et des formes de Jacobi. Nous discutons des applications de ces formules et comme exemple nous dérivons de manière algorithmique un critère analogue au critère de Tunnell concernant des nombres congruentes.

We give a geometric interpretation of an arithmetic rule to generate explicit formulas for the Fourier coefficients of elliptic modular forms and their associated Jacobi forms. We discuss applications of these formulas and derive as an example a criterion similar to Tunnel's criterion for a number to be a congruent number.

DOI : https://doi.org/10.5802/jtnb.44
Classification : 11F11,  11F12,  11F30,  11F37,  11F67,  11F75,  11G05
Mots clés : modular forms, jacobi forms, periods, special values, modular symbols
@article{JTNB_1991__3_1_93_0,
     author = {Skoruppa, Nils-Peter},
     title = {Heegner cycles, modular forms and jacobi forms},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {93--116},
     publisher = {Universit\'e Bordeaux I},
     volume = {3},
     number = {1},
     year = {1991},
     doi = {10.5802/jtnb.44},
     zbl = {0734.11033},
     mrnumber = {1116103},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.44/}
}
Nils-Peter Skoruppa. Heegner cycles, modular forms and jacobi forms. Journal de Théorie des Nombres de Bordeaux, Tome 3 (1991) no. 1, pp. 93-116. doi : 10.5802/jtnb.44. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.44/

[E-Z] M. Eichler and D. Zagier, The theory of Jacobi forms, Birkhäuser, Boston, 1985. | MR 781735 | Zbl 0554.10018

[G-K-Z] B. Gross, W. Kohnen, D. Zagier, Heegner points and derivatives of L-series, II, Math. Ann. 278 (1987), 497-562. | EuDML 164302 | MR 909238 | Zbl 0641.14013

[H] E. Hecke, Mathematische Werke, Vandenhoeck & Ruprecht, Göttingen, 1959. | MR 104550 | Zbl 0092.00102

[M] J.I. Manin, Parabolic points and zeta functions of modular curves, Izv. Akad. Nauk SSSR 36 (1972), 19-66. | MR 314846 | Zbl 0243.14008

[Sh1] G. Shimura, Sur les intégrales attachées aux formes automorphes, J. Math. Soc. Japan 11 (1959), 291-311. | MR 120372 | Zbl 0090.05503

[Sh] G. Shimura, Introduction to the arithmetic theory of automorphic functions, Iwana mi Shoten and Princeton University Press, Princeton, 1971. | MR 314766 | Zbl 0221.10029

[S1] N-P. Skoruppa, Explicit formulas for the Fourier coefficients of Jacobi and elliptic modular forms, Invent. math. 102 (1990), 501-520. | EuDML 143844 | MR 1074485 | Zbl 0715.11024

[S2] N-P. Skoruppa, Binary quadratic forms and the Fourier coefficients of elliptic and Jacobi modular forms, J. Reine Angew. Math. 411 (1990), 66-95. | EuDML 153268 | MR 1072974 | Zbl 0702.11028

[S3] N-P. Skoruppa, Developments in the theory of Jacobi forms, International Conference Automorphic Functions and their Applications, Khabarovsk, 27 June - 4 July 1988, ed. by N. Kuznetsov, V. Bykovsky, The USSR Academy of Science, Khabarovsk, 1990, pp. 167-185. | MR 1096975 | Zbl 0745.11029

[S-Z] N-P. Skoruppa and D. Zagier, Jacobi forms and a certain space of modular forms, Invent. Math. 94 (1988), 113-146. | EuDML 143620 | MR 958592 | Zbl 0651.10020

[T] J.B. Tunnell, A classical diophantine problem and modular forms of weight 3/2, Invent. Math. 72 (1983), 323-334. | EuDML 143024 | MR 700775 | Zbl 0515.10013