Grothendieck–Katz and Bombieri–Dwork conjectures for differential systems

Let $K$ be a number field and let

$$D:Y'=aY,$$

where $a=(a_{i,j})\in K(x)^{n\times n}$. For almost all primes $p$ of $K$, let $\mathfrak p$ be a prime above $p$, and let $a(\mathfrak p)$ denote the reduction modulo $\mathfrak p$ of the matrix obtained from the $p$-th successive derivative system. A differential system is globally nilpotent if $a(\mathfrak p)$ is nilpotent for almost all primes $\mathfrak p$.

Grothendieck–Katz and Bombieri–Dwork conjectures. The Lie algebra of the differential Galois group of $D$ is minimal with the property that, for almost all primes $\mathfrak p$ of $K$, its reduction modulo $\mathfrak p$ contains the $p$-curvature matrix $a(\mathfrak p)$. Moreover, if $D$ is globally nilpotent, then $D$ arises from geometry.

These conjectures relate the arithmetic of $p$-curvature to the differential Galois group and to geometric origin. The source presents them as conjectural principles; no resolution is given in the supplied text.