Modular form

The classical notion of a modular form is the generic term for a broad class of functions on the upper half plane ( elliptic modular forms ) and their higher-dimensional generalizations (eg, Siegel modular forms ), which is considered in the mathematical branches of function theory and number theory. The modern concept of a modular form is its comprehensive reformulation in terms of the representation theory ( automorphic representations ) and arithmetic geometry (p- adic modular forms ). Classical modular forms are special cases of the so-called automorphic forms.

History

Founder of classical ( purely analytical ) theory of modular forms of the 19th century include Richard Dedekind, Felix Klein, Gotthold Eisenstein and Henri Poincaré. The modern theory of modular forms built in the first half of the twentieth century by Erich Hecke and Carl Ludwig Siegel. Modular forms in terms of the representation theory come from Robert Langlands. p- adic modular forms first appear in Nicholas Katz and Jean -Pierre Serre.

Elliptic modular forms for

It should be

The upper half- plane, that is the set of complex numbers with a positive imaginary part.

For an integer is called a holomorphic or meromorphic function on the upper half-plane is a holomorphic or meromorphic elliptic modular form of weight to the group if they

• The functional equation

• " Holomorphic and meromorphic at infinity " is: This means that the function

Note that it follows from the first condition; is therefore well defined.

Is meromorphic and so is called a module function.

If the function is holomorphic on the upper half plane and at infinity, it means an entire modular form.

Also has a zero at, it is called a peak shape.

Properties

Is always for odd k, so the following statements apply to straight k

The modular forms of weight k form a vector space, as well as the entire modular forms and the peak shapes.

Denoting this vector spaces with and, then:

For the dimension of this vector spaces holds:

Since an isomorphism is given by by by multiplying with the tip shape ( discriminant ) of weight 12, the following applies:

Examples

The simplest examples of entire modular forms of weight k are the so-called Eisenstein series, for a module function, the j- function or absolute invariant and for a tip form the discriminant.