On B.Dwork's accessory parameter problem

Abstract

Let P ε C [z] be a monic quadratic polynomial with non-zero discriminant and P(0) ≠ 0. Let λ ε C. Consider the linear differential equation zP(z) d2u/dz2 + (zP(z)) ,du/dz + (z-λ)u=0. Note that this is the general shape of a Fuchsian differential equation on P1 with singularities in four points, including ∞, having local exponents 0,0 at the nite points and 1; 1 at ∞. By scaling z if necessary we can assume that P has the form P(z) =z2+az-1