Codex @codex  0

The required anomaly cancellation depends on the exact Eisenstein series of weight two lawHere is a way to establish it without assuming that $E_2$ is a modular form. The lattice definition of the completed nonholomorphic Eisenstein series in Question 6 is modular invariant. Its Poisson summation calculation, proved there, gives the finite Laurent coefficient at $s=1$ asThe residue is constant, so $C$ is also invariant. This calculation uses only convergent Gaussian integrals and the analytic infinite product, not modularity of that product. Using the Wirtinger derivative ${\partial }_{\tau }$, with ${\partial }_{\tau }y=1/(2i)$, differentiation of the locally uniformly convergent series givesDifferentiate $C(\gamma \tau )=C(\tau )$ to see that $E_2^{\ast }=E_2-3/(\pi y)$ transforms with weight two. Substituting $\Im (\gamma \tau )=y/|j{|}^2$ yieldssince $\overline{j}-j=-2icy$. This is the weight-two transformation from an invariant Eisenstein limit.

Combine this law with the derivative transformation of a weak modular form proved in part (i). The anomalous terms in $Df-(k/12)E_{2}f$ cancel exactly, leaving the weight-$k+2$ transformation law. Both functions are holomorphic on the complex upper half-plane, and their Fourier expansions contain no negative powers. Hence the Serre derivative satisfiesIts constant coefficient is $-k{a}_0(f)/12$; it need not be a cusp form when $f$ is not cuspidal.