Abstract
Every Siegel modular form has a Fourier-Jacobi expansion. This paper provides various sets of Fourier coefficients whose vanishing implies that the associated cusp form is identically zero. We call such setsestimates because in the Fourier series case, an upper bound for the dimension of the vector space of cusp forms is provided by the cardinality of the set. Our general estimates have, among others, those estimates of Siegel and Eichler as corollaries. In particular, one new corollary of our general estimates appears to be superior for computational purposes to all other known estimates. To illustrate the use of this corollary, we prove the known result that the theta series of the latticesD16+ andE8 ⊕E8 are the same in degreen = 3 by computing just one Fourier coefficient. 1991 Mathematics Subject Classification. 11F46 (11F30, 11F27).
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