Hecke algebra action on Siegel modular forms

This thesis has two parts. In the first part I computed Hecke algebra actions in Siegel Modular forms for genus 2 and 3. First the Satake isomorphism from the Hecke algebra H&parl0;GSp2n&parr0; to the representation ring of the dual group is established. This is done explicitly for n = 2 and n = 3. General formulas are also derived for 2 operators in H&parl0;GSp2n&parr0; . This is done mainly by computations involving Kazhdan-Lustig polynomials. I then apply the relative Satake transform on the Siegel parabolic to compute the number of single cosets and then find explicit single coset representatives for each of the Hecke operators. For each of the Hecke operators I group the single coset representatives and then compute the action for each group. The second part of my thesis consists of a new proof of the mass formula for principally polarized superspecial abelian varieties in characteristic p: X 1#AutX =-1p p+122-g &sqbl0;j=1g&parl0;p j+&parl0;-1&parr0;j&parr0;&sqbr0;˙z -1z-3 &ldots;z1-2g;We reduce this to a mass formula on the inner form of the symplectic group associated to the quaternion algebra ramified at p and infinity, then use the theory of the motive of a reductive group to derive the formula.