Hadamard's maximal determinant problem: Difference between revisions
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An '''eigenform''' (meaning simultaneous Hecke eigenform with modular group SL(2,'''Z''')) is a [[modular form]] which is an [[eigenvector]] for all [[Hecke operator]]s ''T<sub>m</sub>'', ''m'' = 1, 2, 3, …. | |||
Eigenforms fall into the realm of [[number theory]], but can be found in other areas of math and science such as [[analysis]], [[combinatorics]], and [[physics]]. A common example of an eigenform, and the only non-cuspidal eigenforms, are the [[Eisenstein series]]. | |||
== Normalization == | |||
There are two different normalizations for an eigenform (or for a modular form in general). | |||
=== Algebraic normalization === | |||
An eigenform is said to be '''normalized''' when scaled so that the ''q''-coefficient in its [[Fourier series]] is one: | |||
:<math>f = a_0 + q + \sum_{i=2}^\infty a_i q^i</math> | |||
where ''q'' = ''e''<sup>2''πiz''</sup>. As the function ''f'' is also an eigenvector under each Hecke Operator ''T<sub>i</sub>'', it has a corresponding eigenvalue. More specifically ''a''<sub>''i''</sub>, ''i'' ≥ 1 turns out to be the eigenvalue of ''f'' corresponding to the Hecke operator ''T<sub>i</sub>''. In the case of that ''f'' is not a cusp form, the eigenvalues can be given explicitly.<ref>{{cite book | title = Introduction to Elliptic Curves and Modular Forms | author = Neal Koblitz | chapter = III.5}}</ref> | |||
=== Analytic normalization === | |||
As in any [[inner product space]], an eigenform can be normalized with respect to its [[Petersson inner product|inner product]]: | |||
:<math>\langle f, f \rangle = 1\,</math> | |||
== Higher levels == | |||
In the case that the [[modular group]] is not the full SL(2,'''Z'''), there is not a Hecke operator for each ''n'' ∈ '''Z''', and as such the definition of an eigenform is changed accordingly: an eigenform is a modular form which is a simultaneous eigenvector for all Hecke operators that act on the space. | |||
{{reflist}} | |||
{{numtheory-stub}}[[Category:Modular forms|*]] | |||
Latest revision as of 14:30, 14 July 2013
An eigenform (meaning simultaneous Hecke eigenform with modular group SL(2,Z)) is a modular form which is an eigenvector for all Hecke operators Tm, m = 1, 2, 3, ….
Eigenforms fall into the realm of number theory, but can be found in other areas of math and science such as analysis, combinatorics, and physics. A common example of an eigenform, and the only non-cuspidal eigenforms, are the Eisenstein series.
Normalization
There are two different normalizations for an eigenform (or for a modular form in general).
Algebraic normalization
An eigenform is said to be normalized when scaled so that the q-coefficient in its Fourier series is one:
where q = e2πiz. As the function f is also an eigenvector under each Hecke Operator Ti, it has a corresponding eigenvalue. More specifically ai, i ≥ 1 turns out to be the eigenvalue of f corresponding to the Hecke operator Ti. In the case of that f is not a cusp form, the eigenvalues can be given explicitly.[1]
Analytic normalization
As in any inner product space, an eigenform can be normalized with respect to its inner product:
Higher levels
In the case that the modular group is not the full SL(2,Z), there is not a Hecke operator for each n ∈ Z, and as such the definition of an eigenform is changed accordingly: an eigenform is a modular form which is a simultaneous eigenvector for all Hecke operators that act on the space.
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