Arithmetic theory of q-difference equations. The q-analogue of Grothendieck-Katz's conjecture on p-curvatures

Lucia Di Vizio (1)

1 LEP - Laboratoire Émile Picard

Grothendieck's conjecture on p-curvatures predicts that an arithmetic differential equation has a full set of algebraic solutions if and only if its reduction in positive characteristic has a full set of rational solutions for almost all finite places. It is equivalent to Katz's conjectural description of the generic Galois group. In this paper we prove an analogous statement for arithmetic q-difference equation.

Domaines: Théorie des nombres [math.NT]; Algèbres quantiques [math.QA]

Lucia Di Vizio. Arithmetic theory of q-difference equations. The q-analogue of Grothendieck-Katz's conjecture on p-curvatures. Inventiones Mathematicae, 2002, 150, pp.517-578. ⟨hal-00021535⟩