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Concept
Cusp form
Summary
In number theory, a branch of mathematics, a cusp form is a particular kind of modular form with a zero constant coefficient in the Fourier series expansion.
Introduction
A cusp form is distinguished in the case of modular forms for the modular group by the vanishing of the constant coefficient a0 in the Fourier series expansion (see q-expansion)
:\sum a_n q^n.
This Fourier expansion exists as a consequence of the presence in the modular group's action on the upper half-plane via the transformation
:z\mapsto z+1.
For other groups, there may be some translation through several units, in which case the Fourier expansion is in terms of a different parameter. In all cases, though, the limit as q → 0 is the limit in the upper half-plane as the imaginary part of z → ∞. Taking the quotient by the modular group, this limit corresponds to a cusp of a modular curve (in the sense of a point added for compactification). So, the definition amounts to saying
Official source
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