# 2021-04-18

## Notes on modular forms

• Modular forms can be defined as functions $${\mathbb{H}}\to {\mathbb{C}}$$ 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 $$H \cup{\mathbb{P}}^1({\mathbb{Q}})$$
• This definition extends to $$H/\Gamma$$
• 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
• $${\mathbb{H}}_S^g/{\mathsf{Sp}}(2g; {\mathbb{Z}})$$ 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.
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