and , then we let
Fix an integer . For and a function defined on , we define
For a finite index subgroup of containing a congruence subgroup, a function defined on is said to be a weight modular form if:
is holomorphic on .
is holomorphic at the cusps.
This last condition requires some explanation. First observe that the element
and , while if satisfies all the other conditions above, . In other words, is periodic with period . Thus, convergence permitting, admits a Fourier expansion. Therefore, we say that is holomorphic at the cusps if, for all , admits a a Fourier expansion
If all the are zero for , then a modular form is said to be a cusp form. The set of modular forms for (respectively cusp forms for ) is often denoted by (respectively ). Both and are finite dimensional vector spaces.
The space of modular forms for (respectively cusp forms) is non-trivial for any even and greater than 4 (respectively greater than and not ). Examples of modular forms for are:
The Weierstrass function, also called the modular discriminant, is a modular form of weight :
Every modular form is expressible as
where the are arbitrary constants, and . Cusp forms are the forms with .
|Date of creation||2013-03-22 14:07:37|
|Last modified on||2013-03-22 14:07:37|
|Last modified by||olivierfouquetx (2421)|