#### Vol. 15, No. 8, 2021

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Universal abelian variety and Siegel modular forms

### Shouhei Ma

Vol. 15 (2021), No. 8, 2089–2122
##### Abstract

We prove that the ring of Siegel modular forms of weight divisible by $g+n+1$ is isomorphic to the ring of (log) pluricanonical forms on the $n$-fold Kuga family of abelian varieties and certain compactifications of it, for every arithmetic group for a symplectic form of rank $2g>2$. We also give applications to the Kodaira dimension of the Kuga variety. In most cases, the Kuga variety has canonical singularities.

##### Keywords
Siegel modular forms, Kuga family, toroidal compactification, Kodaira dimension
##### Mathematical Subject Classification
Primary: 11F46, 14K10