Quasi-modular forms attached to elliptic curves, I

Hossein Movasati[1]

  • [1] Instituto de Matemática Pura e Aplicada, IMPA Estrada Dona Castorina, 110 22460-320, Rio de Janeiro, RJ Brazil

Annales mathématiques Blaise Pascal (2012)

  • Volume: 19, Issue: 2, page 307-377
  • ISSN: 1259-1734

Abstract

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In the present text we give a geometric interpretation of quasi-modular forms using moduli of elliptic curves with marked elements in their de Rham cohomologies. In this way differential equations of modular and quasi-modular forms are interpreted as vector fields on such moduli spaces and they can be calculated from the Gauss-Manin connection of the corresponding universal family of elliptic curves. For the full modular group such a differential equation is calculated and it turns out to be the Ramanujan differential equation between Eisenstein series. We also explain the notion of period map constructed from elliptic integrals. This turns out to be the bridge between the algebraic notion of a quasi-modular form and the one as a holomorphic function on the upper half plane. In this way we also get another interpretation, essentially due to Halphen, of the Ramanujan differential equation in terms of hypergeometric functions. The interpretation of quasi-modular forms as sections of jet bundles and some related enumerative problems are also presented.

How to cite

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Movasati, Hossein. "Quasi-modular forms attached to elliptic curves, I." Annales mathématiques Blaise Pascal 19.2 (2012): 307-377. <http://eudml.org/doc/251068>.

@article{Movasati2012,
abstract = {In the present text we give a geometric interpretation of quasi-modular forms using moduli of elliptic curves with marked elements in their de Rham cohomologies. In this way differential equations of modular and quasi-modular forms are interpreted as vector fields on such moduli spaces and they can be calculated from the Gauss-Manin connection of the corresponding universal family of elliptic curves. For the full modular group such a differential equation is calculated and it turns out to be the Ramanujan differential equation between Eisenstein series. We also explain the notion of period map constructed from elliptic integrals. This turns out to be the bridge between the algebraic notion of a quasi-modular form and the one as a holomorphic function on the upper half plane. In this way we also get another interpretation, essentially due to Halphen, of the Ramanujan differential equation in terms of hypergeometric functions. The interpretation of quasi-modular forms as sections of jet bundles and some related enumerative problems are also presented.},
affiliation = {Instituto de Matemática Pura e Aplicada, IMPA Estrada Dona Castorina, 110 22460-320, Rio de Janeiro, RJ Brazil},
author = {Movasati, Hossein},
journal = {Annales mathématiques Blaise Pascal},
keywords = {Quasimodular forms; modular forms; elliptic curves; Gauss-Manin connections},
language = {eng},
month = {7},
number = {2},
pages = {307-377},
publisher = {Annales mathématiques Blaise Pascal},
title = {Quasi-modular forms attached to elliptic curves, I},
url = {http://eudml.org/doc/251068},
volume = {19},
year = {2012},
}

TY - JOUR
AU - Movasati, Hossein
TI - Quasi-modular forms attached to elliptic curves, I
JO - Annales mathématiques Blaise Pascal
DA - 2012/7//
PB - Annales mathématiques Blaise Pascal
VL - 19
IS - 2
SP - 307
EP - 377
AB - In the present text we give a geometric interpretation of quasi-modular forms using moduli of elliptic curves with marked elements in their de Rham cohomologies. In this way differential equations of modular and quasi-modular forms are interpreted as vector fields on such moduli spaces and they can be calculated from the Gauss-Manin connection of the corresponding universal family of elliptic curves. For the full modular group such a differential equation is calculated and it turns out to be the Ramanujan differential equation between Eisenstein series. We also explain the notion of period map constructed from elliptic integrals. This turns out to be the bridge between the algebraic notion of a quasi-modular form and the one as a holomorphic function on the upper half plane. In this way we also get another interpretation, essentially due to Halphen, of the Ramanujan differential equation in terms of hypergeometric functions. The interpretation of quasi-modular forms as sections of jet bundles and some related enumerative problems are also presented.
LA - eng
KW - Quasimodular forms; modular forms; elliptic curves; Gauss-Manin connections
UR - http://eudml.org/doc/251068
ER -

References

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