Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 137 / 4

Let $\mathcal{L}$ be the set of lattices in $\mathbb{C}$. The lattice model of a modular form isFor the normalization used here, the $n$th Hecke operator on lattice functions isThere are finitely many such superlattices, and scaling gives a bijection between those above $\Lambda$ and those above $\lambda \Lambda$, so $T_{n}F$ again has weight $k$.

For $\tau \in \mathfrak{h}$, put ${\Lambda }_{\tau }=\mathbb{Z}\tau +\mathbb{Z}$ and defineIf $\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in \Gamma (1)$, thenso $f_F(\gamma \tau )=(c\tau +d{)}^k{f}_F(\tau )$. Conversely, if $f$ has this invariance and $\Lambda =\mathbb{Z}{\omega }_1+\mathbb{Z}{\omega }_2$ with $\mathrm{Im}({\omega }_1/{\omega }_2)>0$, defineThe modular transformation law makes this independent of the oriented basis. These constructions are inverse, identifying $V_k$ with $W_k$.

Transporting $T_n$ through this identification givesThis defines the Hecke operator on $W_k$.

Write $\tau =x+iy$ and $\sigma =\mathrm{Re}s>1$. The positive-definite quadratic formis bounded below by $C_{\tau }(m^2+n^2)$ for some $C_{\tau }>0$. Henceand the last two-dimensional lattice sum converges for $\sigma >1$. Thus the nonholomorphic Eisenstein series converges absolutely.

For $\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in \Gamma (1)$,andThe map $(m,n)\mapsto (ma+nc,mb+nd)$ permutes ${\mathbb{Z}}^2\setminus \{0\}$, so absolute convergence permits reindexing and gives $G(\gamma \tau ,s)=G(\tau ,s)$. Hence $G(\tau ,s)\in W_0$.

At weight zero the convention from part (a) isPartition the summands of $G$ according to divisibility of the first integer coordinate by $p$. Directly from the definition,In the sum over $b$, the congruence $n\equiv mb(\mathrm{mod}p)$ has one solution when $p\nmid m$, has $p$ solutions when $p$ divides both $m$ and $n$, and has none when $p\mid m$ but $p\nmid n$. Thereforewhere the contribution from pairs divisible by $p$ was rescaled by $p^{-2s}$. Dividing by $p$ proves the Hecke eigenvalue of a nonholomorphic Eisenstein series: