Notes on 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 \(H/\Gamma\)
  • 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.
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