Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 24 / 5

Here being modular of weight $k$ means obeying the weight transformation law, or equivalently being invariant under the slash operator for modular forms:This transformation property alone does not impose holomorphy; a holomorphic modular form additionally satisfies interior and cusp holomorphy. That distinction is needed for the nonholomorphic completion in the next part.

Put $F(\tau )=f(N\tau )$. For $\gamma \in {\Gamma }_0(N)$, conjugation by $\mathrm{diag}(N,1)$ gives the integral determinant-one matrixSince the denominator in the latter action is $(c/N)(N\tau )+d=c\tau +d$, the full-group modularity of $f$ givesThus argument dilation lowers the group to the Gamma 0 congruence subgroup while retaining the weight. If $f$ is a holomorphic modular form, cusp holomorphy is preserved too: an integral matrix $\mathrm{diag}(N,1)\gamma$ of determinant $N$ factors as an integral determinant-one matrix times an upper triangular positive-determinant matrix. The corresponding slash translate is a constant times $f((a\tau +b)/d)$ with positive $a/d$, whose expansion at infinity has no negative exponents. This checks every rational cusp; for a cusp form its constant term remains zero. It is the level-one instance of the degeneracy map for cusp forms.

Logarithmically differentiate the convergent product of the modular discriminant. On compact subsets of $\mathbb{H}$, absolute convergence permits termwise differentiation and rearrangement of the resulting divisor sums:The division is valid because $\Delta$ has no interior zeros. Its weight-twelve inversion law is $\Delta (-1/\tau )={\tau }^{12}\Delta (\tau )$. Taking a logarithmic derivative, including the derivative $1/{\tau }^2$ of $-1/\tau$, yieldsThus the Eisenstein series of weight two has the anomalous transformation

Let $y=\mathrm{Im}\tau$. Under inversion, $y$ becomes $y/|\tau {|}^2$. Using ${\tau }^2-|\tau {|}^2=2iy\tau$, the anomalous term cancels:The Fourier expansion gives $E_2(\tau +1)=E_2(\tau )$, and translation leaves $y$ unchanged. Since translation and inversion generate the projective modular group and weight two is unchanged by $-I$, the almost holomorphic weight-two Eisenstein series satisfiesIt is modular in the transformation sense, but is not a holomorphic modular form. Substituting $\mathrm{Im}(\gamma \tau )=y/|c\tau +d{|}^2$ into this law also gives the useful full-group anomaly

Write $F(\tau )={\sum }_{M\mid N}c(M)E_2(M\tau )$ and $C={\sum }_{M\mid N}c(M)/M$. For $\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in {\Gamma }_0(N)$, each divisor $M$ of $N$ also divides the lower-left entry $c$. ThusThe full transformation of the Eisenstein series of weight two consequently givesSumming with the prescribed coefficients provesIf $C=0$, this is exactly the desired weight-two law. Conversely, test it at $\gamma =\left(\begin{array}{cc}1& 0\\ N& 1\end{array}\right)\in {\Gamma }_0(N)$. Its anomaly factor $6N(N\tau +1)/(\pi i)$ is nonzero throughout the upper half-plane; if $F$ is modular it forces $C=0$. This proves cancellation of weight-two Eisenstein anomalies:Equivalently the inverse-height terms cancel in $\sum c(M)E_2^{\ast }(M\tau )$, leaving precisely the holomorphic function $F$. Eliminating $c(1)$ shows that all such combinations are spanned by $E_2(\tau )-M{E}_2(M\tau )$ for $M>1$ dividing $N$.