Any thoughts on the following simple example:
There is no algebraic invariant associated to $F(x,y) = ax + by$, by which I mean there is no single-valued function of $(a,b)$ left invariant after applying linear transformations of $(x,y)$ to $F$, however if we restrict to linear transformations using integers there is an 'arithmetical invariant' $G(a,b) = \sum_{m,n} \frac{1}{(am+bn)^k},2<k,(m,n)\neq (0,0)$. This arithmetical invariant to $ax+by$ is what a Weierstrass function is to an elliptic curve.