Editing Modular form (section)

===Line bundles===
The situation can be profitably compared to that which arises in the search for functions on the [[projective space]] P(''V''): in that setting, one would ideally like functions ''F'' on the vector space ''V'' which are polynomial in the coordinates of ''v''&nbsp;≠&nbsp;0 in ''V'' and satisfy the equation ''F''(''cv'')&nbsp;=&nbsp;''F''(''v'') for all non-zero ''c''. Unfortunately, the only such functions are constants. If we allow denominators (rational functions instead of polynomials), we can let ''F'' be the ratio of two [[homogeneous function|homogeneous]] polynomials of the same degree. Alternatively, we can stick with polynomials and loosen the dependence on ''c'', letting ''F''(''cv'')&nbsp;=&nbsp;''c''<sup>''k''</sup>''F''(''v''). The solutions are then the homogeneous polynomials of degree {{mvar|k}}. On the one hand, these form a finite dimensional vector space for each&nbsp;''k'', and on the other, if we let ''k'' vary, we can find the numerators and denominators for constructing all the rational functions which are really functions on the underlying projective space&nbsp;P(''V'').

One might ask, since the homogeneous polynomials are not really functions on P(''V''), what are they, geometrically speaking? The [[algebraic geometry|algebro-geometric]] answer is that they are ''sections'' of a [[sheaf (mathematics)|sheaf]] (one could also say a [[vector bundle|line bundle]] in this case). The situation with modular forms is precisely analogous.

Modular forms can also be profitably approached from this geometric direction, as sections of line bundles on the moduli space of elliptic curves.