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Eigenform

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In mathematics, 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. Another example is the Δ function.

Normalization

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There are two different normalizations for an eigenform (or for a modular form in general).

Algebraic normalization

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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 when f is not a cusp form, the eigenvalues can be given explicitly.[1]

Analytic normalization

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An eigenform which is cuspidal can be normalized with respect to its inner product:

Existence

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The existence of eigenforms is a nontrivial result, but does come directly from the fact that the Hecke algebra is commutative.

Higher levels

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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.

In cybernetics

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In cybernetics, the notion of an eigenform is understood as an example of a reflexive system. It plays an important role in the work of Heinz von Foerster,[2] and is "inextricably linked with second order cybernetics".[3]

References

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  1. ^ Neal Koblitz (1984). "III.5". Introduction to Elliptic Curves and Modular Forms. ISBN 9780387960296.
  2. ^ Foerster, H. von (1981). Objects: tokens for (eigen-) behaviors. In Observing Systems (pp. 274 - 285). The Systems Inquiry Series. Seaside, CA: Intersystems Publications.
  3. ^ Kauffman, L. H. (2003). Eigenforms: Objects as tokens for eigenbehaviors. Cybernetics and Human Knowing, 10(3/4), 73-90.