Eigenform

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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 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[edit]

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

Algebraic normalization[edit]

An eigenform is said to be normalized when scaled so that the q-coefficient in its Fourier series is one:

f = a_0 + q + \sum_{i=2}^\infty a_i q^i

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[edit]

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

\langle f, f \rangle = 1\,

Existence[edit]

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

Higher levels[edit]

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.

References[edit]

  1. ^ Neal Koblitz. "III.5". Introduction to Elliptic Curves and Modular Forms.