mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 diophantine equation, integer solution to an equation by using the Chinese remainder theorem to piece together solutions mo ...

for

linear ordinary differential equation In mathematics, a linear differential equation is a differential equation that is linear in the unknown function and its derivatives, so it can be written in the form a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b(x) where and are arbi ...

s, related to

differential Galois theory In mathematics, differential Galois theory is the field that studies extensions of differential fields. Whereas algebraic Galois theory studies extensions of field (mathematics), algebraic fields, differential Galois theory studies extensions of ...

and in a loose sense analogous to the result in the

Chebotarev density theorem The Chebotarev 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 id ...

considered as the

polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

case. It is a

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...

Alexander Grothendieck Alexander Grothendieck, later Alexandre Grothendieck in French (; ; ; 28 March 1928 – 13 November 2014), was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. His research ext ...

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 topological manifold, ''n''-manifold, the equivalence classes being connected, injective function, injectively immersed submanifolds, all of the same dimension ...

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''-by-''n'' matrix ''A'' of

algebraic function In mathematics, an algebraic function is a function that can be defined as the root of an irreducible polynomial equation. Algebraic functions are often algebraic expressions using a finite number of terms, involving only the algebraic operati ...

s with

algebraic number In mathematics, an algebraic number is a number that is a root of a function, root of a non-zero polynomial in one variable with integer (or, equivalently, Rational number, rational) coefficients. For example, the golden ratio (1 + \sqrt)/2 is ...

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 In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix w ...

). 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 the mathematical theory of special functions, Schwarz's list or the Schwarz table is the list of 15 cases found by when hypergeometric functions can be expressed algebraically. More precisely, it is a listing of parameters determining the case ...

. In

monodromy In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As the name implies, the fundamental meaning of ''mono ...

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 In mathematics, the term "almost all" means "all but a negligible quantity". More precisely, if X is a set (mathematics), set, "almost all elements of X" means "all elements of X but those in a negligible set, negligible subset of X". The meaning o ...

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 Nicholas Michael Katz (; born December 7, 1943) is an American mathematician, working in arithmetic geometry, particularly on ''p''-adic methods, monodromy and moduli problems, and number theory. He is currently a professor of Mathematics a ...

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 generalise the category of linear representations of a ...

techniques to show that this conjecture is essentially the same as saying that the differential Galois group ''G'' (or strictly speaking the

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...

g of the

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

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

Zariski closure In algebraic geometry and commutative algebra, the Zariski topology is a topology defined on geometric objects called varieties. It is very different from topologies that are commonly used in real or complex analysis; in particular, it is not ...

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 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 equation In mathematics, the Picard–Fuchs equation, named after Émile Picard and Lazarus Fuchs, is a linear ordinary differential equation whose solutions describe the periods of elliptic curves. Definition Let :j=\frac be the j-invariant with g_2 and ...

s (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 ...

), 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 theor ...

s. Other progress has been by arithmetic methods.

History

Nicholas Katz related some cases to

deformation theory In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution ''P'' of a problem to slightly different solutions ''P''ε, where ε is a small number, or a vector of small quantities. The infinitesima ...

in 1972, in a paper where the conjecture was published. Since then, reformulations have been published. A q-analogue for

difference equation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...

s 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. 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 introducing items with similar properties as expressions of the form x^n / n! have, also when it is not possible to actually divide by n!. Definition Let '' ...

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

Notes

References