The limiting zero counting measure for Eisenstein series

Abstract

For an arbitrary congruence group $Γ$, we study the zeros of a generating set of the space of Eisenstein series of varying even weight for all $Γ(1)$-conjugates of $Γ$. These zeros form a $Γ(1)$-invariant set that descends to the modular orbifold $Y(1)$. It is known that, as the (even) weight tends to infinity, the set of zeros converges in the Hausdorff metric to finitely many geodesic segments. In this paper, we find the weak limit of the normalised counting measure of the zeros in the (natural) orbifold sense: if a geodesic occurring in the Hausdorff limit is parametrised by the unit circle $S^1$ via a Möbius transformation then the limit measure on this segment is the pullback of the Haar measure on $S^1$ by that transformation. For the proof, we first show a version of the Poincaré--Lelong theorem for orbifold curves, and then apply it on $Y(1)$ to reduce the problem to computing the scaled limit of the subharmonic functions $\log |E|$ associated to Eisenstein series $E$. This turns out to be the maximum of the negative logarithm of the modulus of finitely many linear functions supported on the relevant geodesic segments, and we finish the proof by showing that the associated measure has no atoms and making explicit, on each open segment, the distributional Laplacian of this function in terms of the argument measure. The non-atomicity allows us to conclude a sublinear growth rate in the weight for the total order of all Eisenstein series at a given point in $Y(1)$, and weak convergence also of the non-orbifold measure to the same limit.