Joint IAS/Princeton University Number Theory Seminar

The Grothendieck–Katz $p$-curvature conjecture predicts that an arithmetic differential equation whose reduction modulo $p$ has vanishing $p$-curvatures for almost all $p$, has finite monodromy. It is known that it suffices to prove the conjecture for differential equations on the projective line minus three points. In this talk, I will first focus on this case and introduce a $p$-adic convergence condition, which would hold if the $p$-curvature is defined and vanishes. Using the algebraicity criteria established by André, Bost, and Chambert-Loir, I will prove a variant of this conjecture for the projective line minus three points, which asserts that if the equation satisfies the above convergence condition for all $p$, then its monodromy is trivial. I will also prove a similar variant of the $p$-curvature conjecture for an elliptic curve with $j$-invariant 1728 minus its identity point.