The general case remains unsolved, despite recent progress; it has been linked to geometric investigations involving algebraic foliations.
Then a necessary condition is that for almost all prime numbers p, the system defined by reduction modulo p should also have a full set of algebraic solutions, over the finite field with p elements.
[2] A wide class of cases has been proved by Benson Farb and Mark Kisin;[3] these equations are on a locally symmetric variety X subject to some group-theoretic conditions.
This work is based on the previous results of Katz for Picard–Fuchs equations (in the contemporary sense of the Gauss–Manin connection), as amplified in the Tannakian direction by André.
[6] In responding to Kisin's talk on this work at the 2009 Colloque Grothendieck,[7] Katz gave a brief account from personal knowledge of the genesis of the conjecture.