Gelfond–Schneider theorem: Difference between revisions

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In [[number theory]], a branch of [[mathematics]], a '''cusp form''' is a particular kind of [[modular form]], distinguished in the case of modular forms for the [[modular group]] by the vanishing in the [[Fourier series]] expansion (see [[q-expansion|''q''-expansion]])


:<math>\Sigma a_n q^n</math>


of the constant coefficient ''a<sub>0</sub>''. This Fourier expansion exists as a consequence of the presence in the modular group's action on the [[upper half-plane]] of the transformation
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:<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'' &rarr; 0 is the limit in the upper half-plane as the [[imaginary part]] of ''z'' &rarr; &infin;. Taking the quotient by the modular group, say, 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.
 
The dimensions of spaces of cusp forms are in principle computable, via the [[Riemann-Roch theorem]].  For example, the famous Ramanujan function &tau;(''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'''
 
:&Delta;(''z'', ''q''),
 
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
 
:&tau;(''n'')
 
and called '[[Tau-function|Ramanujan's tau function]]', with the normalization :&tau;(1) = 1.
 
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 subgroup]]s, and corresponding [[cuspidal representation]]s.
 
==References==
*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
*Shimura, Goro, '''An Introduction to the Arithmetic Theory of Automorphic Functions''', [[Princeton University Press]], 1994. ISBN 0-691-08092-5
*Gelbart, Stephen, '''Automorphic Forms on Adele Groups''', Annals of Mathematics Studies, No. 83, Princeton University Press, 1975. ISBN 0-691-08156-5
 
[[Category:Modular forms]]
 
[[ta:இராமானுசன் கணிதத்துளிகள்: டௌ-சார்பு]]

Latest revision as of 09:16, 27 September 2014


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