--- date: 2021-08-05 tags: - web/quick-notes created: 2021-10-27T19:35 updated: 2024-04-19T16:19 --- Tags: #arithmetic-geometry/modular-forms #AG/moduli-spaces #higher-algebra/stacks # 2021-08-05 ## Classical / Analytic Moduli Theory Tags: #projects/notes/reading Refs: [modular form](modular%20form.md) > Reference: see - $\SL_2(\RR)\actson \HH$ transitively by linear fractional transformations, and $\Stab(i) = \SO(2)$. Thus one can realize $\HH \cong \SL_2(\RR)/\SO_2(\RR)$. - Applying a homothety to a [lattice](lattice.md) $\Lambda$ yields $L_\tau \da \ZZ + \ZZ\tau$ for some $\tau\in\HH$ and $\Lambda \cong L_\tau$. Writing an elliptic curve as $\CC/L_\tau$, the moduli of elliptic curves is given by \[ A_1\da \SL_2(\ZZ)\diagdown\HH \cong \dcoset{\SL_2(\RR)}{\SL_2(\ZZ)}{\SO_2(\RR)} .\] This quotient is Hausdorff, and $A_1 \mapsvia{\sim} \CC$ 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 $X(1) \da \overline{A_1}$. This point is referred to as a [cusp](cusp). - $-I$ acts trivially on $\HH$, so this factors through $\Gamma \da \PSL_2(\ZZ) \da \SL_2(\ZZ)/\gens{\pm I}$. - 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: ![](attachments/2021-08-06_01-03-00.png) - $i$ has [isotropy](isotropy) $\gens{S}$, $\zeta_3$ has $\gens{ST}$, and $\zeta_3^2$ has $\gens{TS}$. Applying $S$ and $T^{\pm 1}$ to the fundamental domain $F$ tiles $\HH$ by hyperbolic triangles. - $\PSL_2(\ZZ) = \gens{S, T}$. - Maps $f: A_1\to \CC$ are continuous iff their pullbacks along $\pi: \HH\to A^1$ are continuous, so these are necessarily $\Gamma\dash$invariant functions. - $f$ is [automorphic](automorphic) with automorphy factor $\phi_\gamma(z)$ iff \[ f(\gamma z) = \phi_\gamma(z) f(z) .\] 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 \[ f(z) = (cz+d)^{-2k} f(\gamma z), \gamma \da \matt a b c d .\] - Note that \[ \dd{}{z} {az+b \over cz +d }= {ad-bc \over (cz+d)^2} = {1\over (cz+d)^2} ,\] and so a meromorphic form $\omega = f(z) \dz$ transforms under $\gamma$ as \[ \gamma \cdot f(z)\dz = f(\gamma \cdot z) d(\gamma\cdot z) = (cz+d)^{-2}f(\gamma\cdot z)\dz .\] - So weakly modular forms of weight $2k$ are those form which $\omega^{\tensor k}$ is invariant. - Dropping the *weakly* adjective involves imposing holomorphy conditions at $\infty$. $f$ is a (standard) **[modular form](modular%20form.md) of [weight](weight%20of%20a%20modular%20form) $2k$** if $f$ is weakly modular, holomorphic on $\HH$, and the Fourier coefficients satisfy $a_{<0} = 0$. - $f$ is a [cusp form](cusp%20form.md) if $a_0 = 0$. - Poisson summation: if $f:\RR\to \CC$ is a [Schwartz function](Schwartz%20function) (smooth and super-polynomial decay), then \[ \sum_{n \in \mathbb{Z}} f(n)=\sum_{n \in \mathbb{Z}} \widehat{f}(n) .\] - $\zeta(s)$ is the $L\dash$function associated to the trivial Galois representation \[ \rho_{\Triv}: G_\QQ\to \CC\units .\] $L\dash$functions coming from arbitrary 1-dim reps will correspond to Dirichlet characters by Kronecker-Weber, and are referred to as Dirichlet $L\dash$functions. - $\Gamma(1) \da \SL_2(\ZZ)$, and **principal congruence subgroups of level $N$** for $\Gamma(1)$ are defined as \[ \Gamma(N) \da \ker\qty{\Gamma(1) \surjects \SL_2(\ZZ/N)} = \ts{M\in \Gamma(1) \st M\cong I \mod N} ,\] so the kernels of reduction mod $N$. [Unsorted/songruence subgroups](Unsorted/songruence%20subgroups.md) are any subgroups $H$ such that $\Gamma(N) \subseteq H \leq \Gamma(1)$ for some $N$. - Letting $\Gamma(N)$ act on $\HH$ or $\HH^*$, one can define [modular curves](modular%20curves) \[ X(\Gamma) &\da \Gamma\diagdown \HH^* \\ Y(\Gamma) &\da \Gamma\diagdown \HH ,\] where $\HH^* = \HH \union (\QQ \union \ts{\infty}) \subset \PP^1(\CC)$. - Note that $Y(1)$ parameterizes elliptic curves. - The inclusions $\Gamma \injects \Gamma(1)$ induce a branched cover $X(\Gamma) \surjects X(\Gamma(1)) = A^1 \cong \PP^1(\CC)$. - The genera of these curves can be computed using [Riemann-Hurwitz](Riemann-Hurwitz) : \[ 2 g(Y)-2=(2 g(X)-2) d+\sum_{y \in Y}\left(e_{y}-1\right) ,\] yielding for $N\geq 3$, $g(X(\Gamma(N)))$ is given by \[ g=1+\frac{d(N-6)}{12 N} \qtext{where} d=\frac{1}{2}[\Gamma(1): \Gamma(N)]=\frac{N^{3}}{2} \prod\left(1-\frac{1}{p^{2}}\right) .\] - For $X$ a smooth curve and $D\in \Div(X)$, set \[ \Omega^1_X(D) \da \Omega^1_X \tensor \OO_X(D) \cong \OO_X(\omega + D) \] where $\omega$ is the canonical divisor. Then $\globsec{X; \Omega^1_X(D)}$ is the space of meromorphic 1-forms $\omega$ such that $\Div(\omega) + D \geq 0$ is effective. - Define $M_{2k}(\Gamma)$ to be the space of weight $2k$ modular forms, and $S_{2k}$ the space of cusp forms. Then \( \bigoplus_k S_{2k} \in \gr^\ZZ \Alg\slice{\CC}^\fg \), and for $\SL_2(\ZZ)$ this algebra is generated by the Eisenstein series $G_4$ and $G_6$. - A contravariant functor $F$ admits a [fine moduli space](fine%20moduli%20space.md) $\B F$ if $F$ is representable by $\B F$, i.e. $F(\wait) \cong \Hom(\wait, \B F)$. By Yoneda, $F$ admits a universal family $\E F \to \B F$ so that $F(X)$ is the pullback of it under some map $X\to \B F$. - The functor $F(\wait)$ sending $X$ to isomorphism classes of elliptic curves over $X$ admits $Y(1)$ as a [coarse moduli space](fine%20moduli%20space.md) and not a fine one, since there are nontrivial families with constant $j\dash$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 $H_1(E; \ZZ/N)$, which is symplectic since it carries a pairing with intersection matrix $\matt 0 1 {-1} 0$. - Moduli interpretations: - $Y_1(N) = Y(\Gamma_1(N))$ is a coarse moduli space for pairs $(E, P)$ where $P$ is an $N\dash$torsion point. - $Y_0(N)$ parameterizes $(E, C)$ where $C\leq E[N]$ is a cyclic subgroup of the $N\dash$torsion points. - An [elliptic curve](elliptic%20curve.md) $E$ over a [scheme](scheme.md) $S$ is a [smooth proper morphism](smooth%20proper%20morphism.md) $f:E\to S$ with a section such that the closed fibers of $f$ are genus 1 curves. - Letting $\Ell(S)$ be elliptic curves over $S$ up to isomorphism, $Y(1)\slice{\ZZ} = \spec \ZZ[j]$ is a coarse moduli scheme for $\Ell(\wait)$, and $Y(1) = \qty{Y(1)\slice{\ZZ} \fiberprod{\spec \ZZ} \spec \CC}^\an$ is the associated analytic space. - Look up the [Weil Pairing](Weil%20Pairing) $e_n$. - 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 $N=2$, $Y(2) = \spec \ZZ \adjoin{t, {1\over t(t-1)}}$ as a coarse moduli space, and a corresponding almost-universal family $y^2z = x(x-z)(x-tz)$. - For $N=3$, $Y(2) = \spec R\adjoin{t, {1\over t^3-1}}$ where $R \da \ZZ\adjoin{{1\over 3}, \zeta_3^2}$ 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 $\tv{-1, 0, 1}, [-1, \zeta_3^2, 0]$. ## Moduli as Stacks - Can view an elliptic curve as a pair $(X, p)$ where $X$ is a compact Riemann surface with $\dim_\CC H^0(X; \Omega^1_X) = 1$ 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 $\HH$, forcing it to be constant by Liouville. : - Can define a lattice $\Lambda \subseteq V$ in an arbitrary vector space as a discrete [cocompact](cocompact.md) subgroup, so $V/\Lambda$ is compact. - The **order** of a function $f$ at $x$ is given by \[ \text { ord }_{x}(f):= \begin{cases}0 & \text { if } \mathrm{f} \text { is holomorphic and non-zero at } x \\ k & \text { if } \mathrm{f} \text { has a zero of order } k \text { at } x \\ -k & \text { if } \mathrm{f} \text { has a pole of order } k \text { at } x\end{cases} .\] - Can define divisors as maps $D:X\to \ZZ$ where cofinitely many points are sent to zero. The map $\ord_{\wait}(f)$ is a divisor associated to any function $f$, denoted $(f)$. - Divisors $(f)$ for $f$ meromorphic are principal, and setting $\deg(\sum n_i p_i) \da \sum n_o$, 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 \[ \Phi: H_{1}(X, \mathbb{Z}) & \rightarrow \mathbb{C} \\ \gamma & \mapsto \int_{\gamma} \omega .\] For a fixed nonzero holomorphic 1-form $\omega$, there is a group pf [periods](periods) which forms a lattice over $\CC$: \[ \Lambda \da \ts{\int_\gamma \omega \in \CC \st \gamma \in H_1(X; \ZZ)} = \im\Phi .\] One can recover $(X, P)$ as $(\CC/\Lambda(\omega), 0)$ > To pick back up: