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 | |||
:<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, 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 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''' | |||
:Δ(''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 | |||
:τ(''n'') | |||
and called '[[Tau-function|Ramanujan's tau function]]', with the normalization :τ(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]] | |||
Revision as of 21:05, 13 January 2014
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)
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.
The dimensions of spaces of cusp forms are in principle computable, via the Riemann-Roch theorem. For example, the famous Ramanujan function τ(n) arises as the sequence of Fourier coefficients of the cusp form of weight 12 for the modular group, with a1 = 1. The space of such forms has dimension 1, which means this definition is possible; and that accounts for the action of Hecke operators on the space being by scalar multiplication (Mordell's proof of Ramanujan's identities). Explicitly it is the modular discriminant
- Δ(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
- τ(n)
and called 'Ramanujan's tau function', with the normalization :τ(1) = 1.
In the larger picture of automorphic forms, 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 representations.
References
- Serre, Jean-Pierre, A Course in Arithmetic, Graduate Texts in Mathematics, No. 7, 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