A generalization of Watson transformation and representations of ternary quadratic forms

Let L   be a positive definite (non-classic) ternary ZZ-lattice and let p   be a prime such that a 12Zp-modular component of LpLp is nonzero isotropic and 4⋅dL4⋅dL is not divisible by p. For a nonnegative integer m  , let GL,p(m)GL,p(m) be the genus with discriminant pm⋅dLpm⋅dL on the quadratic space Lpm⊗QLpm⊗Q such that for each lattice T∈GL,p(m)T∈GL,p(m), a 12Zp-modular component of TpTp is nonzero isotropic, and TqTq is isometric to (Lpm)q(Lpm)q for any prime q different from p  . Let r(n,M)r(n,M) be the number of representations of an integer n   by a ZZ-lattice M  . In this article, we show that if m⩽2m⩽2 and n is divisible by p   only when m=2m=2, then for any T∈GL,p(m)T∈GL,p(m), r(n,T)r(n,T) can be written as a linear summation of r(pn,Si)r(pn,Si) and r(p3n,Si)r(p3n,Si) for Si∈GL,p(m+1)Si∈GL,p(m+1) with an extra term in some special case. We provide a simple criterion on when the extra term is necessary, and we compute the extra term explicitly. We also give a recursive relation to compute r(n,T)r(n,T), for any T∈GL,p(m)T∈GL,p(m), by using the number of representations of some integers by lattices in GL,p(m+1)GL,p(m+1) for an arbitrary integer m.