Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 126 / 5 / i

Work initially with $\mathrm{Re}s>1$, where the nonholomorphic Eisenstein series and its differentiated series converge locally absolutely. For $\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in S{L}_2(\mathbb{Z})$, the identitiesshow that each summand at $\gamma \tau$ becomes the summand indexed by $(ma+nc,mb+nd)$ at $\tau$. Reindexing the nonzero integer pairs therefore provesHere $y^s$ and the denominator powers use logarithms of positive real numbers, so there is no ambiguity for complex $s$.

For the nonnegative Laplace-Beltrami operator of the hyperbolic plane, $\Delta y^s=-y^2{\partial }_y^2{y}^s=s(1-s)y^s$. If $(m,n)=r(c,d)$ with $(c,d)$ primitive and $r>0$, choose a matrix in $S{L}_2(\mathbb{Z})$ with bottom row $(c,d)$. The corresponding summand isThe Laplace-Beltrami operator commutes with hyperbolic isometries, so this summand has eigenvalue $s(1-s)$. This also covers $m=0$, or follows there directly. Termwise differentiation now givesThe equation describes an automorphic eigenfunction; it does not assert that this noncuspidal function belongs to the square-integrable discrete spectrum.