The Eichler Commutation Relation And Its Applications

Let k be an odd positive integer,L a lattice on a regular positive definite k-dimensional quadratic space over Q,NL the level of L and M(L)the linear space of θ-series attached to the distinct classes in the genus of L.We prove that,for an odd prime p|NL,if Lp = Lp,丄 Lp,2,where Lp,1 is unimodular,Lp,2 is(p)-modular,and QpLp,2is anisotropic,then M(L;p):=M(L)+Tp2.M(L)is stable under the Hecke operator Tp2.If L2 is isometric to(?)with ε∈Z2x and k:= k-1/2 then M(L;2):＝T22.M(L)+T222.M(L)is stable under the Hecke operator T22.Furthermore,we determine some invariant subspaces of the cusp forms for the Hecke operators.Let f be a positive definite integral ternary quadratic form.Its theta function isθ(z;f)=∑n=0∞ a(n;f)qn.For any fixed square-free positive integer t with a(t;f)≠ 0,we define p(n;t,f):＝ a(tn2;f)/a(t;f).For the case when f = x22 + x22+x32 and t=1,Hurwitz proved that p(n;t,f)is multiplicative and he gave its expression.Cooper and Lam proved four similar formulas and proposed a conjecture for some other cases.Using the results given in this paper,we can check the multiplicative property of p(n;t,f)for many cases.All cases in Cooper and Lam’s conjecture are included in ours.Assume that f is a positive definite integral ternary quadratic form.Let Nf denote the level of f.Assume that there are exactly two classes in gen(f)and let g be a representative for the other class.Assume further that f and g are in the same spinor genus.We show that if(M,Nf)= 1 is an eligible integer which is not square-free,then it can be represented by f.This generalizes Ono and Soundararajan’s result for f ＝ x12+x22+10x32,Pei’s result for f = x12+7x22+7x32 and Kelley’s result for f= x12+x22+7x32.