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 holomorphic function on the complex upper half-plane, \mathcal, that roughly satisfies a functional equation with respect to the group action of the modular group and a growth condition. The theory of modul ...

which is an

eigenvector In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by ...

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 rep ...

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

number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

, 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 as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...

, and

physics Physics is the scientific study of matter, its Elementary particle, 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 whi ...

. 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 generalize ...

. Another example is the Δ function.

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 an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems ...

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 pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, ofte ...

: :

\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 is commutative.

Higher levels

In the case that the

modular group In mathematics, the modular group is the projective special linear group \operatorname(2,\mathbb Z) of 2\times 2 matrices with integer coefficients and determinant 1, such that the matrices A and -A are identified. The modular group acts on ...

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

{{reflist Modular forms