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Geodesic clustering of zeros of Eisenstein series for congruence groups

Sebastián Carrillo Santana, Gunther Cornelissen, Berend Ringeling

TL;DR

We study zeros of Eisenstein series for congruence groups by using a spanning family $\{E_k^{a,b}\}$ and analyzing their Γ(1)-conjugate zeros in the standard fundamental domain. The authors derive an Im-bound $\mathrm{Im}(z)<\kappa_Γ+O(1/k)$, prove that zeros converge in Hausdorff distance to a finite set of geodesic segments as weight grows, and classify algebraic zeros via CM theory with explicit discriminant bounds. For principal groups $\Gamma(N)$, the limit configuration is the arc pair $\mathscr A\cup\mathscr A'$ when $N\not\equiv 0\pmod 4$, with additional geodesic pieces arising when $4|N$, and odd levels yield angular equidistribution of zeros along certain arcs. The work connects Kluyver sums generalizing Ramanujan sums to the geometry of zero sets, yielding quantitative zero-density and speed results and revealing a deep link between analytic truncations, hyperbolic geometry, and CM arithmetic in the zeros of Eisenstein series.

Abstract

We consider a set of generators for the space of Eisenstein series of even weight $k$ for any congruence group $Γ$ and study the set of all of their zeros taken for $Γ(1)$-conjugates of $Γ$ in the standard fundamental domain for $Γ(1)$. We describe (a) an upper bound $κ_Γ+ O(1/k)$ 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 $Γ=Γ(N)$ with $4 \nmid N$, $κ_Γ=1$ and the zeros tend to the unit circle, whereas if $4 \mid N$, $κ_Γ=2$ 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 $\exp(2π\mathrm{i}/3)$ for sufficiently large $k$. For $Γ(N)$ 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 $k \rightarrow \infty$.

Geodesic clustering of zeros of Eisenstein series for congruence groups

TL;DR

We study zeros of Eisenstein series for congruence groups by using a spanning family and analyzing their Γ(1)-conjugate zeros in the standard fundamental domain. The authors derive an Im-bound , prove that zeros converge in Hausdorff distance to a finite set of geodesic segments as weight grows, and classify algebraic zeros via CM theory with explicit discriminant bounds. For principal groups , the limit configuration is the arc pair when , with additional geodesic pieces arising when , and odd levels yield angular equidistribution of zeros along certain arcs. The work connects Kluyver sums generalizing Ramanujan sums to the geometry of zero sets, yielding quantitative zero-density and speed results and revealing a deep link between analytic truncations, hyperbolic geometry, and CM arithmetic in the zeros of Eisenstein series.

Abstract

We consider a set of generators for the space of Eisenstein series of even weight for any congruence group and study the set of all of their zeros taken for -conjugates of in the standard fundamental domain for . We describe (a) an upper bound for their imaginary part; (b) a finite configuration of geodesics segments to which all zeros converge in Hausdorff distance as ; (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 with , and the zeros tend to the unit circle, whereas if , and the limit configuration includes parts of vertical geodesics and circles of radius . In both cases, the only algebraic zeros are at and for sufficiently large . For with 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 .

Paper Structure

This paper contains 35 sections, 55 theorems, 251 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries on Eisenstein series for $\Gamma(N)$
  3. The space of Eisenstein series
  4. Fourier expansion, relations
  5. The norm modular form
  6. Kluyver sums
  7. Definitions and examples
  8. (Non-)vanishing of Kluyver sums
  9. General congruence groups
  10. The trace modular forms $E_{k,\Gamma}^{a,b}$
  11. Kluyver sums for congruence groups
  12. Constant Fourier coefficients and non-vanishing of trace Eisenstein series
  13. Constant Fourier coefficients for specific congruence groups
  14. Downward imaginary concentration
  15. Main term, remainder, constant term
  16. ...and 20 more sections

Key Result

Theorem 1

For $\Gamma$ a congruence group of level $N$, $\kappa_\Gamma$ is finite; in fact, $\kappa_\Gamma \leqslant 2 h_\Gamma$ and $\kappa_\Gamma$ divides $N$. For an even weight $k \geqslant 4$, we have an upper bound where the implied constant is independent of the weight $k$. The bound in imup is optimal, in the sense that there exist zeros $z_i \in Z_{\Gamma}$ ($i=1,\dots$) such that $\lim \mathrm{Im

Figures (8)

  • Figure 1: Every dot represents one of the more than 1000 zeros of all Eisenstein series for $\Gamma(N)$ of even weight in $[4,K]$ in the closure of $\mathcal{F}$, where darker points correspond to higher weight. On the left, $N=3, K=80$; on the right, $N=8, K=60$.
  • Figure 2: Geodesic completion of $\overline Z_{\infty,\Gamma(N)}$ (dashed lines), fundamental domain $\mathcal{F}$ (shaded region), and geodesic segments forming the limit $\overline Z_{\infty,\Gamma(N)}$ of the zeros in the closure of $\mathcal{F}$ as the weight tends to infinity (solid lines); left: $N$ not divisible by $4$, right: $N$ divisible by $4$. These theoretical pictures should be compared with the data from Figure \ref{['manyzeros']}.
  • Figure 3: The six circles related to the limit configuration of zeros for $\Gamma(N)$ with $N$ not divisible by $4$.
  • Figure 4: Solutions of the 'truncation' $1 + 200/z^{20} - 10/(z + 1)^{20} + 10/(z - 1)^{20} = 0$ (dots), superimposed on the limit configuration (solid lines)
  • Figure 5: For $k=94$, $N=10^8+1$, $A=\lfloor N/4\rfloor$, and $B=A-10^3$, the dark blue points represent the zeros of $\mathscr{M}_k^{a,b}(z)$ (inside $\mathcal{F}_1$ we should think of this as an accurate approximation of the actual zeros of $E_k^{a,b}(z)$), the gray points represent the zeros of $f_1(z)$, and the magenta points represent the zeros of $f_2(z)$. When $z$ is close to $C_1$ and $C_2$, the zeros of $\mathscr{M}_k(z)$ are close to the zeros of $w_1/z^k+w_2/(z+1)^k$, which all lie on the third circle that appears in the figure.
  • ...and 3 more figures

Theorems & Definitions (143)

  • Definition
  • Theorem 1
  • Theorem 2
  • Proposition 3
  • Proposition 4
  • Theorem 5
  • Proposition 6
  • Theorem 7
  • Proposition 8
  • Remark
  • ...and 133 more