Hecke operators for Siegel modular forms
by
Jim Hafner
IBM Research
Coauthors: Lynne Walling (University of Colorado, Boulder)

A degree n Siegel modular form F over SP_n(Z) typically has its Fourier expansion written as a sum over symmetric even positive semi-definite integral n×n matrices T. By viewing T instead as a quadratic form on a rank n Z-lattice we write the form F as

F(\tau) = å cls(\Lambda)  a(\Lambda)e*{\Lambda \tau}

where the sum is over isometry classes of lattices \Lambda, a(\Lambda)=a(T), a Fourier coefficient for F for some matrix T representing the quadratic form on \Lambda, and e^*{\Lambda\tau} is a sum of exponential functions e{T[G]\tau}, G running over an appropriate set of change of bases matrices for \Lambda.

From this point of view, that is, using this language of lattices, we present an algorithm for simultaneously determining the coset representatives giving the action of Hecke operators T(p) and T_j(p²), 0 <= j <= n, and also describing the action of these operators on Fourier coefficients in terms of lattices. From here, we develop some relations between operators (e.g., we easily derive an explicit relation between T(p)² and the T_j(p²)). In the case n=2, we use these relations to relate local values of the Fourier coefficients and the Hecke eigenvalues via two-variable generating functions. The coefficients of the generating functions depend explicitly on the Hecke eigenvalues. The parameters (roots) of the generating functions are related explicitly to the Satake parameters.

Date received: March 14, 2000