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cusp form (Definition)

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 $\Delta$ function, also called modular discriminant is a weight 12 cusp form for the full modular group $\sldeuxz$ \begin{equation} \Delta(z)=q\underset{n=1}{\overset{\infty}{\prod}}(1-q^n)^{24} \end{equation} 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.




"cusp form" is owned by olivierfouquetx.
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Cross-references: integral, finite, vector space, modular group, weight, modular discriminant, function, scalar product, Eisenstein series, orthogonal, cusp, coefficient, modular form

This is version 6 of cusp form, born on 2004-01-25, modified 2006-10-08.
Object id is 5536, canonical name is CuspForms2.
Accessed 2348 times total.

Classification:
AMS MSC11F11 (Number theory :: Discontinuous groups and automorphic forms :: Modular forms, one variable)

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