# On the converse to Eisenstein's last theorem

# On the converse to Eisenstein's last theorem

Eisenstein proved, in 1852, that if a function f(z) is algebraic, then its Taylor expansion at a point has coefficients lying in some finitely-generated Z-algebra. I will explain ongoing joint work with Josh Lam which studies the extent to which the converse of this theorem holds. Namely, we conjecture that if f(z) satisfies a (possibly non-linear!) algebraic ODE, non-singular at 0, and its Taylor expansion has coefficients lying in a finitely-generated Z-algebra, then f is algebraic. For linear ODE, we prove this conjecture when f(z) satisfies a Picard-Fuchs equation, with initial conditions the class of an algebraic cycle, and in some other cases. For non-linear ODE, we prove it when f(z) satisfies an "isomonodromy" ODE with "Picard-Fuchs" initial conditions. I'll also discuss some motivic corollaries of these results, and explain their relationship to the Grothendieck-Katz p-curvature conjecture.