Analysis of a Motivic Structure

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Consider the decorated seed: 

$$\mathcal S_{\mathrm dec} = (\mathcal I, \Gamma, \Pi, \chi,\rho,\mathcal L)$$ 

which is a candidate motivic object, with $L$-functions arising through trace constructions on $\mathcal S$. To see what that means, consider:

$$ \mathcal S = (\mathcal I, \Gamma, \Pi) $$

which is a $\Pi$-equivariant $4$-sheeted branched cover 

$$ p: \mathcal I \longrightarrow \mathbf{ \widehat{C}}  $$

equipped with a distinguished embedded graph

$$ \Gamma \subset \mathcal I $$

on the covering surface. $\Gamma$ is combinatorially an octahedral graph. $\Gamma$ contains three distinguished $4$-cycles $\Gamma_x,\Gamma_y, \Gamma_z$ whose union is all of $\Gamma$. The decoration $\chi = \lbrace x,y,z\rbrace$ gives an edge-coloring by $\lbrace x,y,z \rbrace$. This is notably different from Grothendieck's dessins d'enfants, which are downstairs on $\widehat{\mathbf C}$.

Write the holonomy representation

$$ \rho : \pi_1(\Gamma) \longrightarrow U(1) $$

The piecewise mappings $\pi \in \Pi$ (piecewise isometries, polytope exchange transformations in the real case) are given by

$$\pi_i : \mathcal I \to \mathcal I, \quad i \in \{x, y, z\}$$

The action of $\Pi$ on $\mathcal I$ induces automorphisms on $\Gamma$

$$ \pi_{*}: \Gamma \longrightarrow \Gamma $$

which act on the cycle space, in particular on the first homology group

$$ \pi_{*} : H_1(\Gamma, \Bbb Z) \longrightarrow H_1(\Gamma, \Bbb Z).$$

The twisted Ihara zeta function, $\zeta_{\Gamma}(u, \rho)$, sees the cycle space at a coarser level, picking out primitive backtrackless cycles

$$\zeta_\Gamma(u,\rho) = \prod_{[C]} \Bigl(1-\rho(C)\,u^{\ell(C)}\Bigr)^{-1}$$

Since $\Pi$ acts on $\Gamma$, it also acts on $\pi_1(\Gamma)$. 

The holonomy representation updates discretely as $\rho \mapsto \rho\circ \pi_*^{-1}$ and we can study orbits such as

$$\mathcal O_{\zeta}=\left\{ \zeta_\Gamma\!\left(u,\rho\circ \pi_*^{-1}\right) : \pi \in \Pi \right\}.$$

The decoration, $\mathcal L$, is a line bundle over $\Gamma$. It also responds to the action of $\Pi$ and gets twisted.

We can think of $\chi,\rho,\mathcal L$ as datum that are responsible for the twisting, as seen with the holonomy twisted Ihara zeta function. We re-organize the datum as:

$$ \mathcal S_{\mathrm{dec}} = (\mathcal S, \mathscr T) $$

where twisting data is now $\mathscr T = (\chi, \rho, \mathcal L)$.

If we suppress twisting data, we recover zeta functions such as the classical Ihara zeta function as traces over the primitive $\mathcal S = (\mathcal I,\Gamma, \Pi)$. I used the Ihara zeta as the guiding example, but $\Gamma$ is not merely combinatorial. It harbors transport, holonomy, line bundles, around which the associated global $L$-function is organized. 

We may study $\mathcal S$ as a kind of motivic object. I'm interested in finding a natural cohomology theory for $\mathcal S$.

Consider a singular analytic $L^2$ cohomology, which represents the notion that we have a singular space, $\mathcal I$, and we desire a cohomology theory built from analytic objects that are square integrable near the singularities. I'll suggest an ansatz for the singular set of $\mathcal I$, namely that $\Gamma$ and a set of eight cone points comprise $\mathcal I_{\mathrm{sing}}$. 

Take the smooth part $\mathcal I_{\mathrm{reg}}=\mathcal I~\backslash ~\mathcal I_{\mathrm{sing}}$, equip $\mathcal I_{\mathrm{reg}}$ with a metric, and examine differential forms $\omega$ where:

$$ \omega \in L^2, \quad d\omega \in L^2.$$

A first pass model is the $k$-th $L^2$ cohomology of the singular space:

$$  H^k_{(2)}(\mathcal I) = \frac{\lbrace \omega \in L^2\Omega^k(\mathcal I_{\mathrm{reg}}):d\omega = 0\rbrace}{d(L^2\Omega^{k-1}(\mathcal I_{\mathrm{reg}}))} $$

which is defined analytically on the regular locus.

While it's not yet clear to me how to develop the cohomological aspect, we can at least build out the algebro-geometric basis of the structure in question, namely, $\mathcal S$, by defining a surface of revolution in intrinsic coordinates $(u,v) \in I\times S^1$ where we use 

$$ g_\phi = \frac{1}{\phi(u)}du^2 + \phi(u)dv^2, \quad v \sim v+2\pi $$

with $\phi(u)>0$. Since $\lvert g_{\phi}\rvert = 1$, the Laplace-Beltrami operator is:

$$ \Delta = \partial_u(\phi(u)\partial_u)+ \frac{1}{\phi(u)}\partial^2_v $$

Keep in mind that $$\mathcal{I} = \bigcup_{j=1}^{4} \mathcal{O}_j$$

with the sheets $\mathcal{O}_1, \dots, \mathcal{O}_4$ being four spindle orbifolds (topologically, Riemann spheres $\widehat{\mathbf{C}}$ each with two cone points). However in this example we are examining only a single member, say $\mathcal O_1$ not the full object, so the group $\Pi$, does not come into play yet.

A football orbifold has two conical tips with total cone angle $2\pi \alpha$ at each tip $(0<\alpha\le 1$; for a cone of order $q$, $\alpha = 1/q$). Locally near a tip $\phi(u)\sim \alpha^2 r^2$ in a geodesic radius $r$.

Let the azimuthal circle carry a flat $U(1)$ line bundle, with holonomy $e^{2\pi i \varphi}(\varphi \in \Bbb R/\Bbb Z)$. Sections satisfy the twisted periodicity $\Psi(u,v+2\pi)= e^{2\pi i \varphi}\Psi(u,v).$

Fourier-Bloch decomposition gives

$$  \Psi(u,v) = \sum_{m \in \Bbb Z} R_m(u)e^{i(m+\varphi)v}.  $$

So holonomy appears as a shift $m \mapsto m+ \varphi$.

Plugging $\Psi = R_m(u)e^{i(m+\varphi)v}$ into $\Delta\Psi = \lambda \Psi$ yields the Sturm-Liouville problem:

$$ (\phi R'_m)' - \frac{(m+\varphi)^2}{\phi(u)}R_m + \lambda R_m = 0 $$

with regularity at the cone tips.

Near a cone of angle $2\pi \alpha$ one finds Bessel behavior with order $\nu = \frac{|m +\varphi|}{\alpha}$ so the $L^2$ solution behaves like $R_m \sim r^{\nu}$.

If the isotropy at a tip has order $q$, azimuthal modes lie in a fixed coset $q\Bbb Z+r$ for some residue $r \in \lbrace 0,...,q-1\rbrace$. We can encode this as an effective shift

$$ m \in q\Bbb Z + r \quad \iff \quad m + \varphi = qn + (r+ \varphi) \quad (n\in \Bbb Z) $$ 

We define the theta kernel with characteristic $(q,r;\varphi)$

$$ \theta_{q,r;\varphi}(t) = \sum_{n \in \Bbb Z} e^{-\pi(qn+r+\varphi)^2 t} $$

Poisson summation gives the modular inversion

$$ \theta_{q,r;\varphi}(1/t) = \frac{1}{q}t^{-1/2} \sum_{k\in \Bbb Z} \exp\bigg(-\pi \frac{k^2}{q^2 t}\bigg) e^{\frac{2\pi i k}{q}(r+\varphi)}$$

From here we take the Mellin transform with $t^{\frac{s}{2} -1}$ to obtain a functional equation, and then continue it meromorphically to $s \in \Bbb C$.

Forgetting the twisting by setting $q=1$, $r=0$ and $\varphi=0$, we recover the standard completed Riemann factor $\pi^{-s/2}\Gamma(s/2)\zeta(s)$.


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