Epstein Curves: Analytic & Geometric Structures

• Epstein Curves are mathematically defined structures that arise both from the zero landscapes of Epstein zeta functions and as horocycle envelopes in hyperbolic geometry.
• They organize critical and off-critical zeros via continuous curves, exhibiting bifurcation, square-root singularities, and asymptotically equidistant behavior.
• These curves connect analytic and geometric theories, underpinning computational methods and holographic bulk-boundary correspondences relevant to Schwarzian actions.

Epstein curves refer to two distinct, but mathematically related, geometric and analytic structures. The first arises from the study of the loci of nontrivial zeros of Epstein zeta functions, yielding a global “zero-landscape” in the spectral parameter space as relevant shape or dimension parameters vary. The second, rooted in complex analysis and differential geometry, is the envelope in the hyperbolic disk traced by horocycles associated to a circle diffeomorphism, central to recent developments connecting hyperbolic geometry, the Schwarzian action, and holography. Both settings exhibit deep structural and functional analogies, including bifurcation behavior, singularities, and identities relating geometric invariants and analytic functionals.

1. Epstein Curves in the Theory of Epstein Zeta Functions

The $d$-dimensional Epstein zeta function on a hypercubic or rectangular lattice is defined by

$\zeta^{(d)}(s) = \frac12 \sideset{}{'}\sum_{n_1,\ldots,n_d \in \mathbb{Z}} (n_1^2 + \ldots + n_d^2)^{-s/2},\quad \Re(s) > d,$

where the prime excludes the origin. For the two-dimensional rectangular case with spacings $(1, \Delta)$, the function is

$$\zeta^{(2)}(s, \Delta) = \frac12 \sum_{(j, k) \neq (0,0)} (j^2 + \Delta^2 k^2)^{-s}, \quad \Re(s)>1.$$

These functions admit analytic continuation to $\mathbb{C}$ with a simple pole at $s=d$ (or $s=1$ for $d=2$), with explicit representations involving theta functions, functional equations, and symmetries such as $Z(s, \Delta) = Z(1-s, \Delta)$ and $Z(s, \Delta) = Z(s, 1/\Delta)$ (Bétermin et al., 2021, Travěnec et al., 2019).

2. Critical and Off-Critical Zeros: The Emergence of Epstein Curves

Nontrivial zeros $\rho = \rho_x + i\rho_y$ of the Epstein zeta function partition into:

• Critical zeros: $\rho_x = d/2$ (or $1/2$ in $d=2$), i.e., zeros lying on the analog of the Riemann critical line.
• Off-critical zeros: $\rho_x \ne d/2$.

For every fixed lattice shape parameter (e.g., $\Delta$) or dimension $d$, the critical zeros are roots of an explicit real integral equation in $\rho_y$, forming continuous curves $\, \rho_y(\Delta)$ or $\, \rho_y(d)$ as $\Delta$ or $d$ varies. These loci—the Epstein curves (Editor's term)—organize the nontrivial zero-sets in the $(\text{shape/}d, \rho_x, \rho_y)$-space. Off-critical zeros, defined by coupled real equations in $(\rho_x, \rho_y)$, are similarly organized, and new off-critical branches always emanate from special bifurcation points on the critical curves (Bétermin et al., 2021, Travěnec et al., 2019).

3. Edge Points, Square-Root Singularities, and Bifurcation Structure

On each critical Epstein curve there exist isolated "edge points" $(\Delta^*, \rho_y^*)$ (or $(d^*, \rho_y^*)$), where two critical zeros on a branch merge and annihilate. Mathematically, these are identified as points where the tangent $\text{d}\rho_y/\text{d}\Delta$ (or $\text{d}\rho_y/\text{d}d$) diverges, or where the derivative with respect to $\rho_y$ of the corresponding functional vanishes. The local structure near such an edge zero is governed by a square-root singularity: $$\rho_y - \rho_y^* = \pm C \sqrt{|\Delta - \Delta^*|} + \text{higher order terms}, \quad C > 0,$$ with analogous expressions in $d$ (Bétermin et al., 2021, Travěnec et al., 2019).

At these edge points, bifurcation occurs: traversing through an edge zero in parameter space, the real critical zero solution ceases to exist, but a conjugate pair of complex (off-critical) zeros bifurcates off the critical line. The leading order expansion shows that

$$\delta \rho_x \sim \pm \sqrt{-\frac{a}{c}} \sqrt{|\varepsilon|}, \quad \delta \rho_y \sim -\frac{b - ad/c}{2c} \varepsilon,$$

and each off-critical curve connects a right edge point to a left edge point, forming closed paths in the full parameter space (Bétermin et al., 2021).

This bifurcation has direct formal analogy to a mean-field second-order phase transition, where the deviation $\rho_x - d/2$ functions as an order parameter with mean-field exponent $1/2$ (Travěnec et al., 2019).

4. Asymptotic Regimes and Equidistant Zeros

For both the two-dimensional rectangular and $d$-dimensional hypercubic Epstein zeta functions, the spacing and distribution of critical zeros exhibit distinguished asymptotic regimes:

• Small/$\Delta \to 0$ (or $d \to 0$): Using Euler–Maclaurin summation and asymptotics for the zeta and gamma functions, the spacing of critical zeros along the imaginary axis becomes asymptotically equidistant,

$$\rho_y \simeq \frac{\pi n}{|\ln \Delta / \pi|}, \quad n \in \mathbb{Z}^*,$$

tending to zero as $|\ln \Delta| \to \infty$. For $d \to \infty$, the gap is $\sim 2\pi/\ln d$ (Bétermin et al., 2021, Travěnec et al., 2019).

• Real off-critical zeros: For sufficiently small $\Delta < \Delta_c^* \approx 0.141733$, or large $d > d_c^* \approx 9.24555$, there exist real off-critical zeros with $\rho_y=0$ and $\rho_x \to 0$ and $1$ (or $d$) in the respective limits.

Approximate formulas for these zeros are derived from transcendental equations associated with simplified forms of the zeta function in the limiting regime. The value $\Delta_c^*$ is explicitly expressed as $e^{\gamma}/(4\pi)$, $\gamma$ the Euler–Mascheroni constant (Bétermin et al., 2021).

5. Computational Methods and Global Structure

The numerical determination of Epstein curves relies on root-finding for the associated real (and coupled real) integral equations, typically using Mathematica's FindRoot or FindMinimum with high-precision settings. Critical zeros are located to 8–20 digits of accuracy in compute times of order seconds, while continuation methods track off-critical curves emanating from edge points, gluing left and right edge pairs. The union of critical and off-critical Epstein curves provides an explicit charting of the zero locus in three-dimensional parameter space, revealing the structure and transitions among critical, off-critical, and trivial zeros (Bétermin et al., 2021).

6. Epstein Curves as Horocycle Envelopes and Geometric Applications

In hyperbolic geometry, given a $C^3$ circle diffeomorphism $\varphi: S^1 \to S^1$, the Epstein curve $\mathrm{Ep}_h: S^1 \to \mathbb{D}$ in the hyperbolic disk $\mathbb{D}$ is the envelope of the family of horocycles $H_z$ at each boundary point $z = e^{i\theta}$, with horocycle "size" determined by the pullback of the Euclidean metric under $\varphi$. The explicit parametric formula is: $$\mathrm{Ep}_h(e^{i\theta}) = \frac{\sigma_\theta^2 + (e^{2\sigma} - 1)}{\sigma_\theta^2 + (e^{\sigma}+1)^2} e^{i\theta} + \frac{2 \sigma_\theta e^{\sigma}}{\sigma_\theta^2 + (e^{\sigma}+1)^2} i e^{i\theta},$$ with $\sigma(\theta) = \log|\varphi'(e^{i\theta})|$ (Pallete et al., 7 Mar 2025).

This construction yields a natural holographic bulk-boundary correspondence in the disk: the signed hyperbolic length $L(\mathrm{Ep}_h)$ and signed area $A(\mathrm{Ep}_h)$ of the envelope are canonically identified with the Schwarzian action $S[\varphi]$ and its negative, respectively, establishing the identity $S[\varphi] = L(\mathrm{Ep}_h) = -A(\mathrm{Ep}_h)$. This relationship underpins new proofs of the non-negativity of the Schwarzian action and equates isoperimetric defects to the "energy" encoded in Epstein curves.

7. Connections, Extensions, and Physical Interpretations

Epstein curves generalize in both analytic and geometric contexts:

• Higher coadjoint orbits: Analogous horocycle-envelope constructions define Epstein curves with $n$ "punctures," corresponding to orbits $$\mathrm{M\ddot{o}b}_n(S^1)\backslash \mathrm{Diff}(S^1)$$ and Schwarzian actions measuring renormalized areas of null hypersurfaces in $\mathrm{AdS}_3$ with $n$ cusps (Pallete et al., 7 Mar 2025).
• Lorentzian analogues: "Dual" Epstein curves in de Sitter space yield spacelike curves, with Lorentzian area and length identities paralleling those in the hyperbolic case.
• Piecewise and low regularity data: Epstein curves and their invariants extend to $C^{1,1}$ circle homeomorphisms, with added corner terms representing external angles.
• Renormalized geodesic length: Truncating hyperbolic geodesics by boundary horocycles produces a renormalized length exactly equal to the log of the bi-local observable $O(\varphi; u, v)$ in Schwarzian field theory, and the full collection of such observables determines the diffeomorphism up to M\"obius transformations.

Significance: In analysis, number theory, statistical mechanics, and quantum gravity, Epstein curves organize the fine structure of zero sets and encode subtle bifurcation and symmetry phenomena. In geometry and holography, they realize the Hamiltonian and action principles for boundary field theories, with their geometric invariants computing quantities central to the dynamics of diffeomorphism groups, conformal geometry, and lower-dimensional gravity.

Primary sources: (Bétermin et al., 2021, Pallete et al., 7 Mar 2025, Travěnec et al., 2019).