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2002•Compositio MathematicaOpen access

Heegner Divisors and Nonholomorphic Modular Forms

Jens Funke

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Abstract

We consider an embedded modular curve in a locally symmetric space M attached to an orthogonal group of signature (p , 2) and associate to it a nonholomorphic elliptic modular form by integrating a certain theta function over the modular curve. We compute the Fourier expansion and identify the generating series of the (suitably defined) intersection numbers of the Heegner divisors in M with the modular curve as the holomorphic part of the modular form. This recovers and generalizes parts of work of Hirzebruch and Zagier.

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We consider an embedded modular curve in a locally symmetric space M attached to an orthogonal group of signature (p , 2) and associate to it a nonholomorphic elliptic modular form by integrating a certain theta function over the modular curve. We compute the Fourier expansion and identify the generating series of the (suitably defined) intersection numbers of the Heegner divisors in M with the modular curve as the holomorphic part of the modular form. This recovers and generalizes parts of work of Hirzebruch and Zagier.

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Available abstract

We consider an embedded modular curve in a locally symmetric space M attached to an orthogonal group of signature (p , 2) and associate to it a nonholomorphic elliptic modular form by integrating a certain theta function over the modular curve. We compute the Fourier expansion and identify the generating series of the (suitably defined) intersection numbers of the Heegner divisors in M with the modular curve as the holomorphic part of the modular form. This recovers and generalizes parts of work of Hirzebruch and Zagier.

Key concepts: Mathematics, Modular form, Eisenstein series, Modular elliptic curve, Modular curve, Modular design, Pure mathematics, Holomorphic function

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