Are the Igusa invariants (defined in Igusa's oringal paper) also classical Siegel modular forms? I read from somewhere that

$\psi_4=\frac{1}{4}I_4, \quad \psi_6=\frac{1}{8}(I_2I_4-3I_6), \quad \chi_{10}=-\frac{1}{2^{14}}I_{10}, \quad \chi_{12} = \frac{1}{2^{17}3}I_2 I_{10}, \quad \chi_{35} = 5^3 I_{10}^2I_{15}.$ where $\psi_i$ are Eisenstein series and $\chi_i$ cusp forms. So it seems that at least $I_2$ and $I_6$ are modular functions instead of forms. But the puzzle is, where does, for instance, $I_2$ diverge? $I_2$ is also a polynomial of coefficients of the genus 2 curve. So I don't see how it can have poles.




1 Answer 1


Jacobians of genus 2 curves don't give the whole moduli space of 2-dimensional abelian varieties. They miss the locus of products of two elliptic curves (everything with principal polarization). So e.g. $I_2$ should blow up along that locus.


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