In mathematics, an eigenform (meaning simultaneous Hecke eigenform with modular group SL(2,Z)) is a

modular form In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the Group action (mathematics), group action of the modular group, and also satisfying a grow ...

which is an

eigenvector In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...

for all

Hecke operator In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by , is a certain kind of "averaging" operator that plays a significant role in the structure of vector spaces of modular forms and more general automorphic repr ...

s ''T_m'', ''m'' = 1, 2, 3, .... Eigenforms fall into the realm of

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

, but can be found in other areas of math and science such as

analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...

combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...

, and

physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...

. A common example of an eigenform, and the only non-cuspidal eigenforms, are the

Eisenstein series Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modular group, Eisenstein series can be generaliz ...

. Another example is the Δ Function. In

second-order cybernetics Second-order cybernetics, also known as the cybernetics of cybernetics, is the recursive application of cybernetics to itself and the reflexive practice of cybernetics according to such a critique. It is cybernetics where "the role of the observer ...

, eigenforms are an example of a self-referential system.Kauffman, L. H. (2003). Eigenforms: Objects as tokens for eigenbehaviors. Cybernetics and Human Knowing, 10(3/4), 73-90.

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 A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...

is one: :

f = a_0 + q + \sum_^\infty a_i q^i

where ''q'' = ''e''^2''πiz''. As the function ''f'' is also an eigenvector under each Hecke operator ''T_i'', it has a corresponding eigenvalue. More specifically ''a''_''i'', ''i'' ≥ 1 turns out to be the eigenvalue of ''f'' corresponding to the Hecke operator ''T_i''. In the case when ''f'' is not a cusp form, the eigenvalues can be given explicitly.

Analytic normalization

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

inner product In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...

: :

\langle f, f \rangle = 1\,

Existence

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

Hecke algebra In mathematics, the Hecke algebra is the algebra generated by Hecke operators. Properties The algebra is a commutative ring. In the classical elliptic modular form theory, the Hecke operators ''T'n'' with ''n'' coprime to the level acting on ...

is commutative.

Higher levels

In the case that the

modular group In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional l ...

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

cybernetics Cybernetics is a wide-ranging field concerned with circular causality, such as feedback, in regulatory and purposive systems. Cybernetics is named after an example of circular causal feedback, that of steering a ship, where the helmsperson m ...

, 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 Heinz von Foerster (German spelling: Heinz von Förster; November 13, 1911 – October 2, 2002) was an Austrian American scientist combining physics and philosophy, and widely attributed as the originator of Second-order cybernetics. He was twice ...

,Foerster, H. von (1981). Objects: tokens for (eigen-) behaviors. In Observing Systems (pp. 274 - 285). The Systems Inquiry Series. Seaside, CA: Intersystems Publications. and is "inextricably linked with second order cybernetics".

References

{{reflist Modular forms Cybernetics