Arithmetic theory of \(q\)-difference equations. The \(q\)-analogue of Grothendieck-Katz’s conjecture on \(p\)-curvatures.

*(English)*Zbl 1023.12004A famous conjecture of Grothendieck states that a linear differential equation with coefficients that are rational functions over a number field has a full set of algebraic solutions if and only if its reduction modulo almost all primes has a full set of rational solutions over the corresponding prime field; this last statement is in turn equivalent to the vanishing of the so-called \(p\)-curvature, a linear operator defined over the prime field by iteration of the corresponding differential operator.

The conjecture was generalized by Katz in the following form: the Lie algebra of the generic Galois group of a differential equation is the smallest algebraic Lie algebra whose reduction modulo \(p\) contains the \(p\)-curvature for almost all primes \(p\); Katz went on to prove that these conjectures were actually equivalent and that the latter one could be used to compute Galois groups by arithmetic means in many interesting cases. Both conjectures are still open today.

In the paper under review, the author states and proves precise analogs of these conjectures for \(q\)-difference equations. The foremost difference with the differential setting is that reductions must be taken modulo powers of the prime numbers \(p\). The techniques used for the proof are borrowed from the theory of \(G\)-functions. The results are then applied to the effective calculation of some Galois groups. Along with the extension in strengthened form of results of Katz to \(q\)-difference equations in the arithmetic situation, the paper contains some adaptation of \(p\)-adic techniques for differential equations and an analogue of Schwarz’s list for basic hypergeometric series.

The conjecture was generalized by Katz in the following form: the Lie algebra of the generic Galois group of a differential equation is the smallest algebraic Lie algebra whose reduction modulo \(p\) contains the \(p\)-curvature for almost all primes \(p\); Katz went on to prove that these conjectures were actually equivalent and that the latter one could be used to compute Galois groups by arithmetic means in many interesting cases. Both conjectures are still open today.

In the paper under review, the author states and proves precise analogs of these conjectures for \(q\)-difference equations. The foremost difference with the differential setting is that reductions must be taken modulo powers of the prime numbers \(p\). The techniques used for the proof are borrowed from the theory of \(G\)-functions. The results are then applied to the effective calculation of some Galois groups. Along with the extension in strengthened form of results of Katz to \(q\)-difference equations in the arithmetic situation, the paper contains some adaptation of \(p\)-adic techniques for differential equations and an analogue of Schwarz’s list for basic hypergeometric series.

Reviewer: Jacques Sauloy (Toulouse)

##### MSC:

12H99 | Differential and difference algebra |

39A13 | Difference equations, scaling (\(q\)-differences) |

33D15 | Basic hypergeometric functions in one variable, \({}_r\phi_s\) |

11S85 | Other nonanalytic theory |