fix a prime integer $p$. set $r$ the completed valuation ring of the maximal unramified extension of $\mathbb{q}_p$. for $x := x_1(n)$ the modular curve with $n$ at least 4 and coprime to $p$, buium-poonen in 2009 showed that the locus of canonical lifts enjoys finite intersection with preimages of finite rank subgroups of $e(r)$ when $e$ is an elliptic curve with a surjection from $x$. this is done using buium's theory of arithmetic odes, in particular interesting homomorphisms $e(r) \to r$ which are arithmetic analogues of manin maps. \\ \\ in this talk, i will review the general idea behind this result and other applications of arithmetic jet spaces to diophantine geometry and discuss a recent analogous result for quasi-canonical lifts, i.e., those curves with serre-tate parameter a root of unity. here the ode manin maps are insufficient, so we introduce a pde version of buium's theory to provide the needed maps. all of this is joint work with a. buium.