# 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

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topMovasati, 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 -

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