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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 in the Fourier series expansion (see ''q''-expansion) : of the constant coefficient ''a0''. This Fourier expansion exists as a consequence of the presence in the modular group's action on the upper half-plane of the transformation : 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, say, 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 that a cusp form is a modular form that vanishes at a cusp. In the case of other groups, there may be several cusps, and the definition becomes a modular form vanishing at ''all'' cusps. This may involve several expansions. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cusp form」の詳細全文を読む スポンサード リンク
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