# cusp form

A cusp form^{} is a modular form^{} whose first coefficient in any expansion around a cusp is zero. Another more general way to define cusp forms is to consider the forms orthogonal^{} to Eisenstein series^{} with respect to the Petersson scalar product^{}.

The Weierstrass $\mathrm{\Delta}$ function^{}, also called modular discriminant^{} is a weight 12 cusp form for the full modular group ${\text{SL}}_{2}(\mathbb{Z})$

$$\mathrm{\Delta}(z)=q\prod _{n=1}^{\infty}{(1-{q}^{n})}^{24}$$ | (1) |

The vector space^{} of weight $k$ cusp forms for the full modular group is finite dimensionnal and non-trivial for $k$ integral greater than 12 and not 14.

Title | cusp form |
---|---|

Canonical name | CuspForm |

Date of creation | 2013-03-22 14:07:43 |

Last modified on | 2013-03-22 14:07:43 |

Owner | olivierfouquetx (2421) |

Last modified by | olivierfouquetx (2421) |

Numerical id | 9 |

Author | olivierfouquetx (2421) |

Entry type | Definition |

Classification | msc 11F11 |