open access

https://doi.org/10.14943/83959

https://hdl.handle.net/2115/69617

Toward the Siegel ring in genus four

英語

theta functions.

2006

Department of Mathematics, Hokkaido University

Hokkaido University Preprint Series in Mathematics

809

1-23

Runge gave the ring of genus three Siegel modular forms as a quotient ring, 3=hJ(3)i. R3 is the genus three ring of code polynomials and J(3) is the di®erence of he weight enumerators for the e8 © e8 and d+ 6 codes. Freitag and Oura gave a degree 24 elation, R(4) , of the corresponding ideal in genus four; R(4) is also a linear combination of eight enumerators. We take another step toward the ring of Siegel modular forms in genus our. We explain new techniques for computing with Siegel modular forms and actually ompute six new relations, classifying all relations through degree 32. We show that the local odimension of any irreducible component de¯ned by these known relations is at least 3 and hat the true ideal of relations in genus four is not a complete intersection. Also, we explain ow to generate an in¯nite set of relations by symmetrizing ¯rst order theta identities and ive one example in degree 32. We give the generating function of R5 and use it to reprove esults of Nebe [25] and Salvati Manni [37].

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