A modular form is a holomorphic function on the upper half-plane that satisfies certain transformation properties.
Modular forms appear as
An (integral) modular form of weight is a holomorphic function on the upper half-plane
(complex numbers with strictly positive imaginary part)
if acting by we have
(notice that for then )
has at worst a pole at (for weak modular forms this condition is relaxed)
it follows that with is a meromorphic funtion on the open disk.
More abstractly, for the moduli stack of elliptic curves and the corresponding universal bundle, write for the line bundle of fiberwise Kähler differential forms. Write for the 0-section of this line bundle. Then
is a line bundle over the moduli stack of elliptic curves. A modular form of weight is a section of
Links and references
Revised on November 6, 2013 00:53:23
by Urs Schreiber