Rational functions and modular forms.

There are two elementary methods for constructing modular forms that dominate in literature. One of them uses automorphic Poincaré series and the other one theta functions. We start a third elementary approach to modular forms using rational functions that have certain properties regarding pole distribution and growth. We prove modularity with contour integration methods and Weil's converse theorem, without using the classical formalism of Eisenstein series and L-functions. [ABSTRACT FROM AUTHOR]