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.

Classification : 11F11, 11F12, 11F30, 11F37, 11F67, 11F75, 11G05
Mots-clés : modular forms, jacobi forms, periods, special values, modular symbols 
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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. https://jtnb.centre-mersenne.org/item/JTNB_1991__3_1_93_0/

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

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

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

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

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

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

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

[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 | MR | Zbl

[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 | Zbl

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

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