Tags: #modular_forms #moduli_spaces #stacks
2021-08-05
Classical / Analytic Moduli Theory
Tags: #reading_notes Refs: modular form
Reference: see https://www.math.purdue.edu/~arapura/preprints/shimura2.pdf
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\({\operatorname{SL}}_2({\mathbb{R}})\curvearrowright{\mathbb{H}}\) transitively by linear fractional transformations, and \({\operatorname{Stab}}(i) = {\operatorname{SO}}(2)\). Thus one can realize \({\mathbb{H}}\cong {\operatorname{SL}}_2({\mathbb{R}})/{\operatorname{SO}}_2({\mathbb{R}})\).
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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.
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\(-I\) acts trivially on \({\mathbb{H}}\), so this factors through \(\Gamma \coloneqq{\operatorname{PSL}}_2({\mathbb{Z}}) \coloneqq{\operatorname{SL}}_2({\mathbb{Z}})/\left\langle{\pm I}\right\rangle\).
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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:
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\(i\) 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.
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\({\operatorname{PSL}}_2({\mathbb{Z}}) = \left\langle{S, T}\right\rangle\).
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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.
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\(f\) 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 \(g(\gamma z) = g(z)\).
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\(f\) 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*}
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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*}
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So weakly modular forms of weight \(2k\) are those form which \(\omega^{\otimes k}\) is invariant.
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Dropping the weakly adjective involves imposing holomorphy conditions at \(\infty\). \(f\) is a (standard) modular form of [[weight of a modular form|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 \(a_0 = 0\).
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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*}
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\(\zeta(s)\) 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.
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\(\Gamma(1) \coloneqq{\operatorname{SL}}_2({\mathbb{Z}})\), 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 \(N\). Congruence subgroups are any subgroups \(H\) such that \(\Gamma(N) \subseteq H \leq \Gamma(1)\) for some \(N\).
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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.
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The inclusions \(\Gamma \hookrightarrow\Gamma(1)\) induce a branched cover \(X(\Gamma) \twoheadrightarrow X(\Gamma(1)) = A^1 \cong {\mathbb{P}}^1({\mathbb{C}})\).
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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*}
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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.
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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 \(G_6\).
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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 \(X\to {\mathbf{B}}F\).
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The functor \(F({-})\) sending \(X\) to isomorphism classes of elliptic curves over \(X\) admits \(Y(1)\) as a [[fine moduli space|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 \(Y(1)\).
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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 \({ \begin{bmatrix} {0} & {1} \\ {-1} & {0} \end{bmatrix} }\).
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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.
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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.
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Look up the Weil Pairing \(e_n\).
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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.
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For \(N=2\), $Y(2) = \operatorname{Spec}{\mathbb{Z}} { \left[ {t, {1\over t(t-1)}} \right] } $ as a coarse moduli space, and a corresponding almost-universal family \(y^2z = x(x-z)(x-tz)\).
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For \(N=3\), $Y(2) = \operatorname{Spec}R { \left[ {t, {1\over t^3-1}} \right] } $ where $R \coloneqq{\mathbb{Z}} { \left[ {{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
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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. :
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Can define a lattice \(\Lambda \subseteq V\) in an arbitrary vector space as a discrete cocompact subgroup, so \(V/\Lambda\) is compact.
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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*}
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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 \((f)\).
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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.
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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 \(({\mathbb{C}}/\Lambda(\omega), 0)\)
To pick back up: https://repositorio.uniandes.edu.co/bitstream/handle/1992/43725/u830743.pdf?sequence=1