Notes on modular forms
#arithmetic-geometry/modular-forms
-
Modular forms can be defined as functions \({\mathbb{H}}\to {\mathbf{C}}\) satisfying weak \(\Gamma{\hbox{-}}\)invariance.
-
Also sections of a bundle: the modular curve.
-
weight of a modular form : refers to growth rates of these functions.
-
A weight \(k\) modular form is an element of \(H^0(X; \omega^{\otimes k})\) where \(X\) is the compactified modular curve, a quotient of \(H \cup{\mathbf{P}}^1({\mathbf{Q}})\)
- This definition extends to \(H/\Gamma\)
-
A weight \(k\) modular form is an element of \(H^0(X; \omega^{\otimes k})\) where \(X\) is the compactified modular curve, a quotient of \(H \cup{\mathbf{P}}^1({\mathbf{Q}})\)
-
Weird fact: \(M_1\) is one-dimensional, but for \(g\geq 2\) we have \(\dim M_g = 3g-3\)
- Special things for \(g=1\): \(q {\hbox{-}}\)expansions (i.e. Fourier series), vanishing Torelli, \(\pi_1 {\mathbb{T}}\) for the torus is abelian, the \(\theta\) function has a discrete zero locus, infinite product expansions like Jacobi’s triple product
-
Higher genus generalizations come not from a Teichmüller cover \(T_g \twoheadrightarrow M_g\) or \(M_g\), no one seems to care about those though.
- People do care about Siegel modular forms : replace \({\mathbb{H}}\) with \({\mathbb{H}}_S^g\) the symmetric \(g\times g\) matrices with positive-definite imaginary part
-
\({\mathbb{H}}_S^g/{\mathsf{Sp}}(2g; {\mathbf{Z}})\) is somehow a model for moduli stack of abelian varieties, \(M_g\) embeds as a variety since we have the Jacobian
-
Hodge bundle : rank \(g\) over \(M_g\), fibers over isomorphism classes are \(H^0(X, K_X)\) where \(K\) is the canonical bundle, then take the determinant bundle.
- Surprisingly, \({\mathbb{H}}_S^g\) is a Lie group but not a Lie algebra : \([AB]^t = -[BA]^t\), so it’s not closed.