q-expansion principle

In mathematics, the q-expansion principle states that a modular form f has coefficients in a module M if its q-expansion at enough cusps resembles the q-expansion of a modular form g with coefficients in M. It was introduced by Katz (1973, corollaries 1.6.2, 1.12.2).

References

• Katz, Nicholas M. (1973), "p-adic properties of modular schemes and modular forms", Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture Notes in Mathematics, 350, Berlin, New York: Springer-Verlag, pp. 69–190, doi:10.1007/978-3-540-37802-0_3, ISBN 978-3-540-06483-1