# 2021-08-05

## Classical / Analytic Moduli Theory

• $${\operatorname{SL}}_2({\mathbb{R}})\curvearrowright{\mathbb{H}}$$ transitively by linear fractional transformations, and $${\operatorname{Stab}}(i) = {\operatorname{SO}}(2)$$. Thus one can realize $${\mathbb{H}}\cong {\operatorname{SL}}_2({\mathbb{R}})/{\operatorname{SO}}_2({\mathbb{R}})$$.

• Applying a homothety to a lattice $$\Lambda$$ yields $$L_\tau \coloneqq{\mathbb{Z}}+ {\mathbb{Z}}\tau$$ for some $$\tau\in{\mathbb{H}}$$ and $$\Lambda \cong L_\tau$$. Writing an elliptic curve as $${\mathbb{C}}/L_\tau$$, the moduli of elliptic curves is given by \begin{align*} A_1\coloneqq{\operatorname{SL}}_2({\mathbb{Z}})\diagdown{\mathbb{H}}\cong \dcoset{{\operatorname{SL}}_2({\mathbb{R}})}{{\operatorname{SL}}_2({\mathbb{Z}})}{{\operatorname{SO}}_2({\mathbb{R}})} .\end{align*} This quotient is Hausdorff, and $$A_1 \xrightarrow{\sim} {\mathbb{C}}$$ as topological spaces. Somehow this comes from “gluing the two bounding lines of $$F$$ and folding the circular boundary in half,” yielding the sphere minus a point.

• One can naturally compactify this by adding the point at infinity to obtain $$X(1) \coloneqq\overline{A_1}$$. This point is referred to as a cusp.
• $$-I$$ acts trivially on $${\mathbb{H}}$$, so this factors through $$\Gamma \coloneqq{\operatorname{PSL}}_2({\mathbb{Z}}) \coloneqq{\operatorname{SL}}_2({\mathbb{Z}})/\left\langle{\pm I}\right\rangle$$.

• Letting $$S= (z\mapsto -1/z) = \left(\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right), T = (z\mapsto z+1) = \left(\begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\right)$$, there are fundamental domains:

• $$i$$ has isotropy $$\left\langle{S}\right\rangle$$, $$\zeta_3$$ has $$\left\langle{ST}\right\rangle$$, and $$\zeta_3^2$$ has $$\left\langle{TS}\right\rangle$$. Applying $$S$$ and $$T^{\pm 1}$$ to the fundamental domain $$F$$ tiles $${\mathbb{H}}$$ by hyperbolic triangles.

• $${\operatorname{PSL}}_2({\mathbb{Z}}) = \left\langle{S, T}\right\rangle$$.

• Maps $$f: A_1\to {\mathbb{C}}$$ are continuous iff their pullbacks along $$\pi: {\mathbb{H}}\to A^1$$ are continuous, so these are necessarily $$\Gamma{\hbox{-}}$$invariant functions.

• $$f$$ is automorphic with automorphy factor $$\phi_\gamma(z)$$ iff \begin{align*} f(\gamma z) = \phi_\gamma(z) f(z) .\end{align*} For any two such functions, their ratio $$g=f_1/f_2$$ satisfies $$g(\gamma z) = g(z)$$.

• $$f$$ is weakly modular of weight $$2k$$ if \begin{align*} f(z) = (cz+d)^{-2k} f(\gamma z), \gamma \coloneqq{ \begin{bmatrix} {a} & {b} \\ {c} & {d} \end{bmatrix} } .\end{align*}

• Note that \begin{align*} {\frac{\partial }{\partial z}\,} {az+b \over cz +d }= {ad-bc \over (cz+d)^2} = {1\over (cz+d)^2} ,\end{align*} and so a meromorphic form $$\omega = f(z) \,dz$$ transforms under $$\gamma$$ as \begin{align*} \gamma \cdot f(z)\,dz= f(\gamma \cdot z) d(\gamma\cdot z) = (cz+d)^{-2}f(\gamma\cdot z)\,dz .\end{align*}

• So weakly modular forms of weight $$2k$$ are those form which $$\omega^{\otimes k}$$ is invariant.

• Dropping the weakly adjective involves imposing holomorphy conditions at $$\infty$$. $$f$$ is a (standard) modular form of weight $$2k$$ if $$f$$ is weakly modular, holomorphic on $${\mathbb{H}}$$, and the Fourier coefficients satisfy $$a_{<0} = 0$$.

• $$f$$ is a cusp form if $$a_0 = 0$$.
• Poisson summation: if $$f:{\mathbb{R}}\to {\mathbb{C}}$$ is a Schwartz function (smooth and super-polynomial decay), then \begin{align*} \sum_{n \in \mathbb{Z}} f(n)=\sum_{n \in \mathbb{Z}} \widehat{f}(n) .\end{align*}

• $$\zeta(s)$$ is the $$L{\hbox{-}}$$function associated to the trivial Galois representation \begin{align*} \rho_{\mathop{\mathrm{Triv}}}: G_{\mathbb{Q}}\to {\mathbb{C}}^{\times} .\end{align*} $$L{\hbox{-}}$$functions coming from arbitrary 1-dim reps will correspond to Dirichlet characters by Kronecker-Weber, and are referred to as Dirichlet $$L{\hbox{-}}$$functions.

• $$\Gamma(1) \coloneqq{\operatorname{SL}}_2({\mathbb{Z}})$$, and principal congruence subgroups of level $$N$$ for $$\Gamma(1)$$ are defined as \begin{align*} \Gamma(N) \coloneqq\ker\qty{\Gamma(1) \twoheadrightarrow{\operatorname{SL}}_2({\mathbb{Z}}/N)} = \left\{{M\in \Gamma(1) {~\mathrel{\Big\vert}~}M\cong I \operatorname{mod}N}\right\} ,\end{align*} so the kernels of reduction mod $$N$$. Unsorted/songruence subgroups are any subgroups $$H$$ such that $$\Gamma(N) \subseteq H \leq \Gamma(1)$$ for some $$N$$.

• Letting $$\Gamma(N)$$ act on $${\mathbb{H}}$$ or $${\mathbb{H}}^*$$, one can define modular curves \begin{align*} X(\Gamma) &\coloneqq\Gamma\diagdown {\mathbb{H}}^* \\ Y(\Gamma) &\coloneqq\Gamma\diagdown {\mathbb{H}} ,\end{align*} where $${\mathbb{H}}^* = {\mathbb{H}}\cup({\mathbb{Q}}\cup\left\{{\infty}\right\}) \subset {\mathbb{P}}^1({\mathbb{C}})$$.

• Note that $$Y(1)$$ parameterizes elliptic curves.
• The inclusions $$\Gamma \hookrightarrow\Gamma(1)$$ induce a branched cover $$X(\Gamma) \twoheadrightarrow X(\Gamma(1)) = A^1 \cong {\mathbb{P}}^1({\mathbb{C}})$$.

• The genera of these curves can be computed using Riemann-Hurwitz : \begin{align*} 2 g(Y)-2=(2 g(X)-2) d+\sum_{y \in Y}\left(e_{y}-1\right) ,\end{align*} yielding for $$N\geq 3$$, $$g(X(\Gamma(N)))$$ is given by \begin{align*} g=1+\frac{d(N-6)}{12 N} {\quad \operatorname{where} \quad} d=\frac{1}{2}[\Gamma(1): \Gamma(N)]=\frac{N^{3}}{2} \prod\left(1-\frac{1}{p^{2}}\right) .\end{align*}

• For $$X$$ a smooth curve and $$D\in \operatorname{Div}(X)$$, set \begin{align*} \Omega^1_X(D) \coloneqq\Omega^1_X \otimes{\mathcal{O}}_X(D) \cong {\mathcal{O}}_X(\omega + D) \end{align*} where $$\omega$$ is the canonical divisor. Then $${{\Gamma}\qty{X; \Omega^1_X(D)} }$$ is the space of meromorphic 1-forms $$\omega$$ such that $$\operatorname{Div}(\omega) + D \geq 0$$ is effective.

• Define $$M_{2k}(\Gamma)$$ to be the space of weight $$2k$$ modular forms, and $$S_{2k}$$ the space of cusp forms. Then $$\bigoplus_k S_{2k} \in {\mathsf{gr}\,}^{\mathbb{Z}}{\mathsf{Alg}}_{/ {{\mathbb{C}}}} ^{\mathrm{fg}}$$, and for $${\operatorname{SL}}_2({\mathbb{Z}})$$ this algebra is generated by the Eisenstein series $$G_4$$ and $$G_6$$.

• A contravariant functor $$F$$ admits a fine moduli space $${\mathbf{B}}F$$ if $$F$$ is representable by $${\mathbf{B}}F$$, i.e. $$F({-}) \cong \mathop{\mathrm{Hom}}({-}, {\mathbf{B}}F)$$. By Yoneda, $$F$$ admits a universal family $${\mathbf{E}}F \to {\mathbf{B}}F$$ so that $$F(X)$$ is the pullback of it under some map $$X\to {\mathbf{B}}F$$.

• The functor $$F({-})$$ sending $$X$$ to isomorphism classes of elliptic curves over $$X$$ admits $$Y(1)$$ as a coarse moduli space and not a fine one, since there are nontrivial families with constant $$j{\hbox{-}}$$invariant. Despite this, $$E\to j(E)$$ gives a bijection between isomorphism classes of elliptic curves and points of $$Y(1)$$.

• A level $$N$$ structure is a basis for $$H_1(E; {\mathbb{Z}}/N)$$, which is symplectic since it carries a pairing with intersection matrix $${ \begin{bmatrix} {0} & {1} \\ {-1} & {0} \end{bmatrix} }$$.

• Moduli interpretations:

• $$Y_1(N) = Y(\Gamma_1(N))$$ is a coarse moduli space for pairs $$(E, P)$$ where $$P$$ is an $$N{\hbox{-}}$$torsion point.
• $$Y_0(N)$$ parameterizes $$(E, C)$$ where $$C\leq E[N]$$ is a cyclic subgroup of the $$N{\hbox{-}}$$torsion points.
• An elliptic curve $$E$$ over a scheme $$S$$ is a smooth proper morphism $$f:E\to S$$ with a section such that the closed fibers of $$f$$ are genus 1 curves.

• Letting $$\mathrm{Ell} (S)$$ be elliptic curves over $$S$$ up to isomorphism, $$Y(1)_{/ {{\mathbb{Z}}}} = \operatorname{Spec}{\mathbb{Z}}[j]$$ is a coarse moduli scheme for $$\mathrm{Ell} ({-})$$, and $$Y(1) = \qty{Y(1)_{/ {{\mathbb{Z}}}} { \underset{\scriptscriptstyle {\operatorname{Spec}{\mathbb{Z}}} }{\times} } \operatorname{Spec}{\mathbb{C}}}^{\mathrm{an}}$$ is the associated analytic space.
• Look up the Weil Pairing $$e_n$$.

• A level $$N$$ structure is a pair of points $$P, Q \in E[N]$$ generating a subgroup with that $$e_n(P, Q) = \zeta_N$$ is a primitive $$N$$th root of unity. More generally, for curves over schemes, this is a pair of sections inducing level structures on closed fibers.

• For $$N=2$$, $Y(2) = \operatorname{Spec}{\mathbb{Z}} { \left[ \scriptstyle {t, {1\over t(t-1)}} \right] }$ as a coarse moduli space, and a corresponding almost-universal family $$y^2z = x(x-z)(x-tz)$$.

• For $$N=3$$, $Y(2) = \operatorname{Spec}R { \left[ \scriptstyle {t, {1\over t^3-1}} \right] }$ where $R \coloneqq{\mathbb{Z}} { \left[ \scriptstyle {{1\over 3}, \zeta_3^2} \right] }$ for $$\zeta_3$$ a primitive third root of unity. The universal family is $$x^3 + y^3 + z^3 = 3txyz$$, where the level 3 structure is given by the sections $${\left[ {-1, 0, 1} \right]}, [-1, \zeta_3^2, 0]$$.

## Moduli as Stacks

• Can view an elliptic curve as a pair $$(X, p)$$ where $$X$$ is a compact Riemann surface with $$\dim_{\mathbb{C}}H^0(X; \Omega^1_X) = 1$$ and $$p$$ is a point.

• Why elliptic curves have 1-dimensional homology: any globally defined holomorphic 1-form is a double periodic holomorphic 1-form on $${\mathbb{H}}$$, forcing it to be constant by Liouville. :
• Can define a lattice $$\Lambda \subseteq V$$ in an arbitrary vector space as a discrete cocompact subgroup, so $$V/\Lambda$$ is compact.

• The order of a function $$f$$ at $$x$$ is given by

\begin{align*} \text { ord }_{x}(f):= \begin{cases}0 & \text { if } \mathrm{f} \text { is holomorphic and non-zero at } x \\ k & \text { if } \mathrm{f} \text { has a zero of order } k \text { at } x \\ -k & \text { if } \mathrm{f} \text { has a pole of order } k \text { at } x\end{cases} .\end{align*}

• Can define divisors as maps $$D:X\to {\mathbb{Z}}$$ where cofinitely many points are sent to zero. The map $${\operatorname{Ord}}_{{-}}(f)$$ is a divisor associated to any function $$f$$, denoted $$(f)$$.

• Divisors $$(f)$$ for $$f$$ meromorphic are principal, and setting $$\deg(\sum n_i p_i) \coloneqq\sum n_o$$, it turns out that $$\deg((f)) = 0$$ for $$f$$ principal.

• As a consequence, meromorphic functions have equal numbers of zeros and poles, and 1-forms that are not identically zero can not have zeros.
• The period map is defined as \begin{align*} \Phi: H_{1}(X, \mathbb{Z}) & \rightarrow \mathbb{C} \\ \gamma & \mapsto \int_{\gamma} \omega .\end{align*} For a fixed nonzero holomorphic 1-form $$\omega$$, there is a group pf periods which forms a lattice over $${\mathbb{C}}$$: \begin{align*} \Lambda \coloneqq\left\{{\int_\gamma \omega \in {\mathbb{C}}{~\mathrel{\Big\vert}~}\gamma \in H_1(X; {\mathbb{Z}})}\right\} = \operatorname{im}\Phi .\end{align*} One can recover $$(X, P)$$ as $$({\mathbb{C}}/\Lambda(\omega), 0)$$