  We establish Sturm bounds for degree g Siegel modular forms modulo a prime p,
which are vital for explicit computations. Our inductive proof exploits
Fourier-Jacobi expansions of Siegel modular forms and properties of
specializations of Jacobi forms to torsion points. In particular, our approach
is completely different from the proofs of the previously known cases g=1,2,
which do not extend to the case of general g.
