Elliptic Modular Forms and Their Applications - Up To

48 D. Zagier5 Modular Forms and Differential OperatorsThe starting point for this section is the observation that the derivative ofa modular form is not modular, but nearly is. Specifically, if f is a modularform of weight k with the Fourier expansion (3), then by differentiating (2)we see that the derivativeDf = f ′ := 1 df2πi dz = q df∞ dq =∑na n q n (51)(where the factor 2πi has been included in order to preserve the rationalityproperties of the Fourier coefficients) satisfies( ) az + bf ′ cz + d= (cz + d) k+2 f ′ (z) + k2πi c (cz + d)k+1 f(z) . (52)If we had only the first term, then f ′ would be a modular form of weight k +2.The presence of the second term, far from being a problem, makes the theorymuch richer. To deal with it, we will:• modify the differentiation operator so that it preserves modularity;• make combinations of derivatives of modular forms which are again modular;• relax the notion of modularity to include functions satisfying equationslike (52);• differentiate with respect to t(z) rather than z itself, where t(z) is a modularfunction.These four approaches will be discussed in the four subsections 5.1–5.4, respectively.5.1 Derivatives of Modular FormsAs already stated, the first approach is to introduce modifications of the operatorD which do preserve modularity. There are two ways to do this, oneholomorphic and one not. We begin with the holomorphic one. Comparingthe transformation equation (52) with equations (19) and (17), we find thatfor any modular form f ∈ M k (Γ 1 ) the functionn=1ϑ k f := f ′ − k 12 E 2 f, (53)sometimes called the Serre derivative, belongs to M k+2 (Γ 1 ). (We will oftendrop the subscript k, since it must always be the weight of the form to whichthe operator is applied.) A first consequence of this basic fact is the following.We introduce the ring ˜M ∗ (Γ 1 ):=M ∗ (Γ 1 )[E 2 ]=C[E 2 ,E 4 ,E 6 ], called the ringof quasimodular forms on SL(2, Z). (An intrinsic definition of the elements ofthis ring, and a definition for other groups Γ ⊂ G, will be given in the nextsubsection.) Then we have: