In his letter [Ser96], J.-P. Serre proves that the systems of Hecke eigenvalues given by modular forms (mod p) are the same as the ones given by locally constant functions A×B/B × → F̄p, where B is the endomorphism algebra of a supersingular elliptic curve. After giving a detailed exposition of Serre’s result, we prove that the systems of Hecke eigenvalues given by Siegel modular forms (mod p) o...

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