Cusp form

2024-03-22T17:09:02Z

129.64.0.36:

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]] [[Series expansion|expansion]].

==Introduction==
A cusp form is distinguished in the case of modular forms for the [[modular group]] by the vanishing of the constant coefficient ''a''<sub>0</sub> in the [[Fourier series]] expansion (see [[q-expansion|''q''-expansion]])

:<math>\sum a_n q^n.</math>

This Fourier expansion exists as a consequence of the presence in the modular group's action on the [[upper half-plane]] via the transformation

:<math>z\mapsto z+1.</math>

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 (singularity)|cusp]] of a [[modular curve]] (in the sense of a point added for [[compactification (mathematics)|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.

==Dimension==
The dimensions of spaces of cusp forms are, in principle, computable via the [[Riemann–Roch theorem]]. For example, the [[Ramanujan tau function]] ''τ''(''n'') arises as the sequence of Fourier coefficients of the cusp form of weight 12 for the modular group, with ''a''<sub>1</sub> = 1. The space of such forms has dimension 1, which means this definition is possible; and that accounts for the action of [[Hecke operator]]s on the space being by [[scalar multiplication]] (Mordell's proof of Ramanujan's identities). Explicitly it is the '''modular discriminant'''

:<math>\Delta(z,q),</math>

which represents (up to a [[normalizing constant]]) the [[discriminant]] of the cubic on the right side of the [[Weierstrass equation]] of an [[elliptic curve]]; and the 24-th power of the [[Dedekind eta function]]. The Fourier coefficients here are written
<math display="block">\tau(n)</math>
and called '[[Ramanujan tau function|Ramanujan's tau function]]', with the normalization ''τ''(1) = 1.

==Related concepts==
In the larger picture of [[automorphic form]]s, the cusp forms are complementary to [[Eisenstein series]], in a ''discrete spectrum''/''continuous spectrum'', or ''discrete series representation''/''induced representation'' distinction typical in different parts of [[spectral theory]]. That is, Eisenstein series can be 'designed' to take on given values at cusps. There is a large general theory, depending though on the quite intricate theory of parabolic subgroups, and corresponding [[cuspidal representation]]s.

Consider <math>P=MU</math> a standard parabolic subgroup of some reductive group <math>G</math> (over <math>\mathbb{A}</math>, the [[adele ring]]), an automorphic form <math>\phi</math> on <math>U(\mathbb{A})M(k)\backslash G</math> is called cuspidal if for all parabolic subgroups <math>P'</math> such that <math>P_0\subset P'\subsetneq P</math> we have <math>\phi_{P'}=0</math>, where <math>P_0</math> is the standard minimal parabolic subgroup. The notation <math>\phi_{P}</math> for <math>P=MU</math> is defined as <math>\phi_P (g) =\int_{U(k)\backslash U(\mathbb{A})} \phi(ug) du</math>.

==References==
*[[Jean-Pierre Serre|Serre, Jean-Pierre]], ''A Course in Arithmetic'', [[Graduate Texts in Mathematics]], No. 7, [[Springer Science+Business Media|Springer-Verlag]], 1978. {{ISBN|0-387-90040-3}}
*[[Goro Shimura|Shimura, Goro]], ''An Introduction to the Arithmetic Theory of Automorphic Functions'', [[Princeton University Press]], 1994. {{ISBN|0-691-08092-5}}
*[[Stephen Gelbart|Gelbart, Stephen]], ''Automorphic Forms on Adele Groups'', Annals of Mathematics Studies, No. 83, Princeton University Press, 1975. {{ISBN|0-691-08156-5}}
* [[Colette Moeglin|Moeglin C]], [[Jean-Loup Waldspurger|Waldspurger JL]] ''Spectral Decomposition and Eisenstein Series: A Paraphrase of the Scriptures'', Schneps L, trans. [[Cambridge University Press]]; 1995. {{ISBN|978-0521418935}}
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[[Category:Modular forms]]

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Cusp form