Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 137 / 4

The left cosets ${\Gamma }_{\infty }\backslash \Gamma (1)$ correspond to primitive bottom rows $(c,d)\in {\mathbb{Z}}^2$, up to simultaneous sign. PutFor $\tau =x+iy$, the identitiesshow that the absolute value of a summand isThe hypothesis $\mathrm{Re}s>(2-k)/2$ says precisely that $r>2$.

If $\tau$ ranges over a compact subset $K\subset \mathfrak{h}$, the positive-definite quadratic form $|c\tau +d{|}^2$ has a uniform lower boundfor some $C_K>0$. The summands are therefore bounded uniformly on $K$ by a constant timesThe corresponding two-dimensional lattice sum converges for $r>2$. The Weierstrass M-test proves absolute and locally uniform convergence. This is the absolute convergence of a weight-k real-analytic Eisenstein series.

For $\delta \in \Gamma (1)$, right multiplication by $\delta$ permutes ${\Gamma }_{\infty }\backslash \Gamma (1)$. The automorphy factor identitytherefore givesThe modular form $f$ obeys the same weight-$k$ transformation law, whileConsequentlyThus the product is invariant under the weight-zero action of $\Gamma (1)$, as described by the invariant product with a weight-k real-analytic Eisenstein series.