Geodesic clustering of zeros of Eisenstein series for congruence groups

We consider a set of generators for the space of Eisenstein series of even weight $k$ for any congruence group $\mathrm{\Gamma }$ and study the set of all of their zeros taken for $\mathrm{\Gamma }(1)$-conjugates of $\mathrm{\Gamma }$ in the standard fundamental domain for $\mathrm{\Gamma }(1)$. We describe (a) an upper bound ${\kappa }_{\mathrm{\Gamma }}+O(1\mathrm{/}k)$ for their imaginary part; (b) a finite configuration of geodesics segments to which all zeros converge in Hausdorff distance as $k\to \mathrm{\infty }$; © a finite set containing all algebraic zeros for all weights. The bound in (a) depends on the (non-)vanishing of a new generalization of Ramanujan sums. The proof of (b) originates in a method used to study phase transitions in statistical physics. The proof of © relies on the theory of complex multiplication. The results can be made quantitative for specific groups. For $\mathrm{\Gamma }=\mathrm{\Gamma }(N)$ with $4\nmid N$, ${\kappa }_{\mathrm{\Gamma }}=1$ and the zeros tend to the unit circle, whereas if $4\mid N$, ${\kappa }_{\mathrm{\Gamma }}=2$ and the limit configuration includes parts of vertical geodesics and circles of radius $2$. In both cases, the only algebraic zeros are at $\mathrm{i}$ and $\exp (2\pi \mathrm{i}\mathrm{/}3)$ for sufficiently large $k$. For $\mathrm{\Gamma }(N)$ with $N$ odd, we use finer estimates to prove a trichotomy for the exact `convergence speed’ of the zeros to the unit circle, as well as angular equidistribution of the zeros as $k\to \mathrm{\infty }$.