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.