Grothendieck–Katz p-curvature conjecture
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In mathematics, the Grothendieck–Katz p-curvature conjecture is a problem on linear ordinary differential equations, related to differential Galois theory and in a loose sense analogous to the result in the Chebotarev density theorem considered as the polynomial case. It is a conjecture of Alexander Grothendieck from the late 1960s, and apparently not published by him in any form; it has been publicised, reformulated and in some cases related to deformation theory proved by Nick Katz in a series of papers. It remains unsolved in the general case as of 2005, but has been linked to some quite broad geometric investigations involving algebraic foliations.
In a simplest possible statement, in which the p-curvature is not explicit, it can be stated in its essentials for a vector system written as
- dv/dz = A(z)v
for a vector v of size n, and an n×n matrix A of algebraic functions with algebraic number coefficients. The question is to give a criterion for when there is a full set of algebraic function solutions, meaning a fundamental matrix (i.e. n vector solutions put into a block matrix). For example, a classical question was for the hypergeometric equation: when does it have a pair of algebraic solutions, in terms of its parameters? The answer is known classically as Schwartz's list. In monodromy terms, the question is of identifying the cases of finite monodromy group.
By reformulation and passing to a larger system, the essential case is for rational functions in A and rational number coefficients. 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.
Grothendieck's conjecture is that these necessary conditions, for almost all p, should be sufficient. The connection with p-curvature is that the mod p condition stated is the same as saying the p-curvature, formed by an operation on A, is zero; so another way to say it is that p-curvature of 0 for almost all p implies enough algebraic solutions of the original equation.
Katz has applied Tannakian category techniques to show that this conjecture is essentially the same as saying that the differential Galois group G (or strictly speaking the Lie algebra g of the algebraic group G, which in this case is the Zariski closure of the monodromy group) can be determined by mod p information, for a certain wide class of differential equations.
