A Singular Cylinder and the Shadow of a Modular Surface

-

Let $\{ f_t(x) := e^{\frac{t}{\log x}} \}_{t \in [1/2,2]}$ be a smooth family of functions on $(0,1)$, and let $F_t(s)$ denote their Mellin transforms:

$$F_t(s) := \int_0^1 f_t(x) \, x^{s-1} \, dx = 2 \sqrt{\frac{t}{s}} \, K_1(2\sqrt{ts}),$$

where $K_1$ is the modified Bessel function of the second kind. Define the Hermitian metric

$$g_t(s) := |F_t(s)|^2 \, ds \otimes d\bar{s}$$

on the punctured complex plane $\mathbb{C}^\times$.

Let $S$ be the topological quotient obtained by identifying the boundary curves $f_{1/2}(x)$ and $f_2(x')$ along lines of slope $+1$ in the $(x,f)$-plane. Then:

The space $S$ is homeomorphic to the $2$-sphere with two conical singularities, arising from the boundary identifications. The metric $g_t(s)$ induces a globally defined Hermitian metric $g^{\sim}(s)$ on $S$, with conical singularities at $s = 0$ and $s = \infty$. Near $s = 0$, the metric behaves as

    $$g_t(s) \sim \frac{1}{|s|^2} ds \otimes d\bar{s},$$

    which corresponds to the canonical flat cone metric of angle $2\pi$ on $\mathbb{C}^\times$. Near $s \to \infty$, the metric decays exponentially as

    $$g_t(s) \sim e^{-4\sqrt{ts}} \, |ds|^2,$$

producing an exponentially collapsing end. While this does not define a true conical singularity in the sense of angle deficit or curvature concentration, it produces an effective degeneration of the metric volume, allowing the end to be compactified topologically, though not metrically, as a cone point. In other words we have a conical singularity at one pole, and an infinitely thin neck at the other pole (not singular in curvature, but vanishing in volume).    

The surface $S$ may be interpreted as a doubly pinched cylinder: a Hermitian surface conformally equivalent to the open cylinder $S^1 \times \mathbb{R}$, with one genuine conical singularity and one asymptotically collapsing end, induced respectively by Bessel blow-up and decay.

Question

Can the spectral-pinched surface $S$, defined via Mellin–Bessel transforms and slope-aligned boundary identifications, be interpreted as a model for a singular compactification of a modular-like surface, where the conical singularity and collapsing end respectively resemble an elliptic fixed point and a cusp?

On modular curves $\Gamma\backslash\Bbb H \cup \lbrace \mathrm{cusps} \rbrace$ we encounter elliptic fixed points i.e. finite order points with cone angle $2\pi/m$, and cusps i.e. infinite volume ends where Eisenstein series and Poincaré series exhibit exponential decay.


In my surface $S$, the point $s=0$ where $g_t(s)\sim\frac{1}{|s|^2}\vert ds \vert^2$ is a flat cone with angle $2\pi$ - resembling an elliptic point of order $1$. Also we have the point $s\to\infty$ where $F_t(s)\sim e^{-2\sqrt{ts}}$ has metric $g_t(s)\sim e^{-4\sqrt{ts}}\vert ds\vert^2$, which mirrors the metric behavior near a cusp. So, $S$ does resemble a modular curve with one cusp and one elliptic fixed point of order $1$.

Additionally, in automorphic theory, Fourier-Whittaker expansions of Maass forms and Eisenstein series often take a form of a series of modified Bessel functions of the second kind which is similar to the form I'm dealing with.

All of this suggests that there may be more than just an analogy going on, and $S$ could be a kind of modular surface.

Comments

Post a Comment

Popular posts from this blog

An introduction to the $\mathcal F$-completion, the Interface construction and the Gamma set

-
Image
I'll start off with a definition that might initially sound pretty abstract and maybe confusing. The point in defining things as I do is to make our lives easier later on. The goal is to focus on a symmetrical subspace of what I'll call the $\mathcal F$-completion. This subspace is called $\mathcal I$. So let's begin: Let $X$ be a smooth, open, geodesically convex $n$-dimensional manifold with regular polytope boundary, and let $V = \{v_i\}$ be a finite set of vertices in $\partial X$. A block is defined as $\mathcal{B} := X \cup \partial X$, where $\partial X$ denotes the boundary. A foliational completion (or $\mathcal{F}$-completion) of a block $\mathcal{B}$ is a structure obtained by extending foliations to accumulate at specific vertices in the boundary. Formally, for each pair of distinct vertices $(v_i, v_j) \in V \times V$, there exists a foliation $\mathcal{F}_{v_{ij}}$, whose leaves accumulate at $v_i$ and $v_j$. The $\mathcal{F}$-completion is the union of these ...
Read more

Grothendieck Site over the Spectral Tower

-
Sheaf Stack over the Dimensional Stratification Site We define a stratified site $\mathcal{S}$ indexed by dimension $n \in \mathbb{Z}_{\geq 2}$, whose objects are geometric-combinatorial models $\mathcal{I}^{(n)}$ associated to dimension-$n$ F-completions. Morphisms in this site correspond to boundary inclusions, coordinate projections, and degenerations between dimensions: $$\mathcal{I}^{(n)} \to \mathcal{I}^{(n-1)} \quad \text{via coordinate slice or foliation boundary}.$$ Let $\mathcal{M}_\Delta^{(n)}$ denote the moduli space of $\Delta$-twisted holonomy representations in dimension $n$, with $\mathcal{M}_\Delta^{(2)} = \mathrm{U}(1)^r / \Delta$ the classical representation torus quotient. We define a sheaf stack of zeta data over $\mathcal{S}$: $$\mathscr{Z} : \mathcal{S}^{\mathrm{op}} \longrightarrow \mathbf{Stacks}$$ such that: For each $n$, $\mathscr{Z}(\mathcal{I}^{(n)})$ is a stack (e.g. derived or topological) encoding the family of zeta sheaves over $\mathcal{M}_\Delta^{(n)}...
Read more

Automorphisms and zeta space

-
Image
Here I will operate under the assumption that $\phi_s$ is a unique and well-posed solution to the following PDE. This means whatever conditions are needed for that to occur, will be imposed. $$s\frac{\partial^2}{\partial s^2}\phi_s(x)=\mp x\frac{\partial}{\partial x}\phi_s(x) $$ for which we have our pre-assigned unique solution: $$\mathcal \phi_{s}:=\bigg\lbrace e^{\frac{\pm s}{\log x}}:s \in \Bbb R \bigg\rbrace$$ We have singularities at $s=0$ and $x=0,1$ and this will be called a "cross singularity" or  $CS$. A natural idea would be to intialize some distribution on this $CS$ and run time forward giving a wellposed analytic solution for all time. So we choose our pre-assigned solution $\phi_s$ and run the solution back in time where we obtain what we wanted, at least qualitatively. We can throw out our solution for $s<0$ because it doesn't directly apply to the situation, and it does not rapidly decay. Meanwhile, for $s>0$ it is Schwartz i.e. has a rapid decay cr...
Read more