In mathematics, the Grothendieck–Katz p-curvature conjecture is a

local-global principle In mathematics, Helmut Hasse's local–global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of eac ...

for linear ordinary differential equations, related to

differential Galois theory In mathematics, differential Galois theory studies the Galois groups of differential equations. Overview Whereas algebraic Galois theory studies extensions of algebraic fields, differential Galois theory studies extensions of differential field ...

and in a loose sense analogous to the result in the

Chebotarev density theorem Chebotarev's density theorem in algebraic number theory describes statistically the splitting of primes in a given Galois extension ''K'' of the field \mathbb of rational numbers. Generally speaking, a prime integer will factor into several ideal ...

considered as the

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...

case. It is a conjecture of Alexander Grothendieck from the late 1960s, and apparently not published by him in any form. The general case remains unsolved, despite recent progress; it has been linked to geometric investigations involving algebraic

foliation In mathematics (differential geometry), a foliation is an equivalence relation on an ''n''-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension ''p'', modeled on the decomposition of ...

Formulation

In a simplest possible statement the conjecture 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 function In mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms, involving only the algebraic operations additi ...

s 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 Schwarz'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 a recurrence 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's formulation for the Galois group

Nicholas Katz has applied

Tannakian category In mathematics, a Tannakian category is a particular kind of monoidal category ''C'', equipped with some extra structure relative to a given field ''K''. The role of such categories ''C'' is to approximate, in some sense, the category of linear re ...

techniques to show that this conjecture is essentially the same as saying that the

differential Galois group In mathematics, differential Galois theory studies the Galois groups of differential equations. Overview Whereas algebraic Galois theory studies extensions of algebraic fields, differential Galois theory studies extensions of differential field ...

''G'' (or strictly speaking the Lie algebra g of the

algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Ma ...

''G'', which in this case is the

Zariski closure In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is ...

of the monodromy group) can be determined by mod ''p'' information, for a certain wide class of differential equations.

Progress

A wide class of cases has been proved by

Benson Farb Benson Stanley Farb (born October 25, 1967) is an American mathematician at the University of Chicago. His research fields include geometric group theory and low-dimensional topology. Early life A native of Norristown, Pennsylvania, Farb earned ...

and Mark Kisin; 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 In mathematics, the Gauss–Manin connection is a connection on a certain vector bundle over a base space ''S'' of a family of algebraic varieties V_s. The fibers of the vector bundle are the de Rham cohomology groups H^k_(V_s) of the fibers V_s o ...

), as amplified in the Tannakian direction by André. It also applies a version of superrigidity particular to

arithmetic group In mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example \mathrm_2(\Z). They arise naturally in the study of arithmetic properties of quadratic forms and other classical topics in number the ...

s. Other progress has been by arithmetic methods.

History

Nicholas Katz related some cases to deformation theory in 1972, in a paper where the conjecture was published. Since then, reformulations have been published. A

q-analogue In mathematics, a ''q''-analog of a theorem, identity or expression is a generalization involving a new parameter ''q'' that returns the original theorem, identity or expression in the limit as . Typically, mathematicians are interested in ''q' ...

for difference equations has been proposed. In responding to Kisin's talk on this work at the 2009 Colloque Grothendieck,Video record.
/ref> Katz gave a brief account from personal knowledge of the genesis of the conjecture. Grothendieck put it forth in public discussion in Spring 1969, but wrote nothing on the topic. He was led to the idea by foundational intuitions in the area of

crystalline cohomology In mathematics, crystalline cohomology is a Weil cohomology theory for schemes ''X'' over a base field ''k''. Its values ''H'n''(''X''/''W'') are modules over the ring ''W'' of Witt vectors over ''k''. It was introduced by and developed by . ...

, at that time being developed by his student

Pierre Berthelot Pierre Berthelot (; born 1943) is a mathematician at the University of Rennes. He developed crystalline cohomology and rigid cohomology. Publications *Berthelot, Pierre ''Cohomologie cristalline des schémas de caractéristique p>0.'' Lecture ...

. In some way wishing to equate the notion of "nilpotence" in the theory of connections, with the

divided power structure In mathematics, specifically commutative algebra, a divided power structure is a way of making expressions of the form x^n / n! meaningful even when it is not possible to actually divide by n!. Definition Let ''A'' be a commutative ring with an ...

technique that became standard in crystalline theory, Grothendieck produced the conjecture as a by-product.

Notes

References