Heegner cycles, modular forms and jacobi forms

Nils-Peter Skoruppa

Journal de théorie des nombres de Bordeaux (1991)

  • Volume: 3, Issue: 1, page 93-116
  • ISSN: 1246-7405

Abstract

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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.

How to cite

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Skoruppa, Nils-Peter. "Heegner cycles, modular forms and jacobi forms." Journal de théorie des nombres de Bordeaux 3.1 (1991): 93-116. <http://eudml.org/doc/93538>.

@article{Skoruppa1991,
abstract = {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.},
author = {Skoruppa, Nils-Peter},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {modular forms; jacobi forms; periods; special values; modular symbols; explicit basis; elliptic modular forms; Fourier coefficients; Jacobi forms; periods of a modular form; computations; Hecke eigenform; congruent number},
language = {eng},
number = {1},
pages = {93-116},
publisher = {Université Bordeaux I},
title = {Heegner cycles, modular forms and jacobi forms},
url = {http://eudml.org/doc/93538},
volume = {3},
year = {1991},
}

TY - JOUR
AU - Skoruppa, Nils-Peter
TI - Heegner cycles, modular forms and jacobi forms
JO - Journal de théorie des nombres de Bordeaux
PY - 1991
PB - Université Bordeaux I
VL - 3
IS - 1
SP - 93
EP - 116
AB - 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.
LA - eng
KW - modular forms; jacobi forms; periods; special values; modular symbols; explicit basis; elliptic modular forms; Fourier coefficients; Jacobi forms; periods of a modular form; computations; Hecke eigenform; congruent number
UR - http://eudml.org/doc/93538
ER -

References

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  1. [E-Z] M. Eichler and D. Zagier, The theory of Jacobi forms, Birkhäuser, Boston, 1985. Zbl0554.10018MR781735
  2. [G-K-Z] B. Gross, W. Kohnen, D. Zagier, Heegner points and derivatives of L-series, II, Math. Ann.278 (1987), 497-562. Zbl0641.14013MR909238
  3. [H] E. Hecke, Mathematische Werke, Vandenhoeck & Ruprecht, Göttingen, 1959. Zbl0092.00102MR104550
  4. [M] J.I. Manin, Parabolic points and zeta functions of modular curves, Izv. Akad. Nauk SSSR36 (1972), 19-66. Zbl0243.14008MR314846
  5. [Sh1] G. Shimura, Sur les intégrales attachées aux formes automorphes, J. Math. Soc. Japan11 (1959), 291-311. Zbl0090.05503MR120372
  6. [Sh] G. Shimura, Introduction to the arithmetic theory of automorphic functions, Iwana mi Shoten and Princeton University Press, Princeton, 1971. Zbl0221.10029MR314766
  7. [S1] N-P. Skoruppa, Explicit formulas for the Fourier coefficients of Jacobi and elliptic modular forms, Invent. math.102 (1990), 501-520. Zbl0715.11024MR1074485
  8. [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. Zbl0702.11028MR1072974
  9. [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. Zbl0745.11029MR1096975
  10. [S-Z] N-P. Skoruppa and D. Zagier, Jacobi forms and a certain space of modular forms, Invent. Math.94 (1988), 113-146. Zbl0651.10020MR958592
  11. [T] J.B. Tunnell, A classical diophantine problem and modular forms of weight 3/2, Invent. Math.72 (1983), 323-334. Zbl0515.10013MR700775

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