Geodesic clustering of zeros of Eisenstein series for congruence groups

Abstract

We consider a set of generators for the space of Eisenstein series of even weight $k$ for any congruence group $\Gamma$ and study the set of all of their zeros taken for $\Gamma(1)$-conjugates of $\Gamma$ in the standard fundamental domain for $\Gamma(1)$. We describe (a) an upper bound $\kappa_\Gamma + O(1/k)$ for their imaginary part; (b) a finite configuration of geodesics segments to which all zeros converge in Hausdorff distance as $k \rightarrow \infty$; (c) 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 (c) relies on the theory of complex multiplication. The results can be made quantitative for specific groups. For $\Gamma=\Gamma(N)$ with $4 \nmid N$, $\kappa_\Gamma=1$ and the zeros tend to the unit circle, whereas if $4 \mid N$, $\kappa_\Gamma=2$ 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 $\exp(2\pi \mathrm{i}/3)$ for sufficiently large $k$. For $\Gamma(N)$ 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 $k \rightarrow \infty$.