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An analogue of Stokes phenomenon for q -difference equations Jacques Sauloy Generalities An analogue of Stokes phenomenon for Slopes q -difference equations Classification Jacques Sauloy Institut Math´ ematique de Toulouse Inria-Rocquencourt, 4 juin 2012

An analogue of Contents Stokes phenomenon for q -difference equations Jacques Sauloy Generalities Slopes Generalities on q -difference equations, systems and modules Classification The slope filtration Local analytic classification of irregular equations

An analogue of Stokes phenomenon for q -difference equations Jacques Sauloy Abstract Generalities In a common work 1 with Jean-Pierre Ramis and Changgui Zhang, Slopes we described an analogue of the Stokes phenomenon for linear Classification analytic complex q-difference equations and used it to get the local analytic classification. If time permits, I will also show how it was applied in a common work with J.-P. R. the Galois theory of such equations. 1 Accepted for publication by Ast´ erisque; meanwhile, see URL http://front.math.ucdavis.edu/0903.0853.

Origin An analogue of Stokes The program of analytic classification of q -difference equations phenomenon for q -difference was first proposed and realized by Birkhoff in 1913 in the context equations of a unified treatment of the Riemann-Hilbert correspondence for Jacques Sauloy fuchsian differential, difference and q -difference equations. The classification program was extended by Birkhoff and Guenter in Generalities 1941 for irregular equations, but never pursued: Slopes Classification “Up to the present time, the theory of linear q -difference equations has lagged noticeably behind the sister theories of linear difference and differential equations. In the opinion of the autors, the use of the canonical system, as formulated above in a special case, is destined to carry the theory of q -difference equations to a comparable degree of completeness. This program includes in particular the complete theory of convergence and divergence of formal series, the explicit determination of the essential transcendental invariants (constants in the canonical form), the inverse Riemann theory both for the neighborhood of x = ∞ and in the complete plane (case of rational coefficients), explicit integral representation of the solutions, and finally the definition of q-sigma periodic matrices, so far defined essentially only in the case n = 1. Because of its extensiveness this material cannot be presented here.” G.D. Birkhoff, 1941

An analogue of Plan Stokes phenomenon for q -difference equations Jacques Sauloy Generalities Slopes Generalities on q -difference equations, systems and modules Classification The slope filtration Local analytic classification of irregular equations

An analogue of Generalities Stokes phenomenon for General notations q -difference equations q ∈ C , | q | > 1. Jacques Sauloy For f ∈ K := C ( { z } ) or f ∈ ˆ K := C (( z )): Generalities σ q f ( z ) := f ( qz ) . Slopes Classification A (complex analytic) linear q -difference equation writes: f ( q n z ) + a 1 ( z ) f ( q n − 1 z ) + · · · + a n ( z ) f ( z ) = 0 , where a 1 , . . . , a n ∈ K , a n � = 0. Encoding: Lf = 0 , q + a 1 σ n − 1 σ n where L := + · · · + a n ∈ D q , K , q σ q , σ − 1 � � D q , K := K , (Ore ring) , q and a 1 , . . . , a n ∈ K , a n � = 0 . Formal equation: replace K by ˆ K .

An analogue of Generalities Stokes phenomenon for Equations, systems, q -difference modules q -difference equations By vectorialisation the q -difference equation Lf = 0 can be Jacques Sauloy turned into a q -difference system: Generalities Slopes   f Classification . . σ q X = AX , A ∈ GL n ( K ) , where X =  ,   .  σ n − 1 f q then into a q -difference module M = ( E , Φ) , with E := K n , Φ := Φ A : X �→ A − 1 σ q X . (Compare with vector spaces equipped with a connection.) Equivalently, M is a left D q , K -module of finite length. Formal equation: replace K by ˆ K .

An analogue of Generalities Stokes phenomenon for Analytic, formal classification q -difference equations Jacques Sauloy Generalities Morphisms from ( K n , Φ A ) to ( K n , Φ B ) correspond to Slopes matrices F ∈ GL n ( K ) such that ( σ q F ) A = BF . Classification Thus, if Y = FX , then σ q X = AX ⇒ σ q Y = BY . Local analytic classification: we say that A ∼ B if there exists a gauge transformation F ∈ GL n ( K ) such that: B = F [ A ] := ( σ q F ) AF − 1 . Formal classification: the same with F ∈ GL n ( ˆ K ).

An analogue of Generalities Stokes phenomenon for Newton polygon (at 0) q -difference equations The q -difference operator P has a Newton polygon at 0, Jacques Sauloy which consists in slopes µ 1 < · · · < µ k ∈ Q together with their multiplicities r 1 , . . . , r k ∈ N ∗ . (Precise definition Generalities omitted !) Slopes Classification By the cyclic vector lemma, any q -difference module can be written M = D q , K / D q , K P . Theorem and definition The Newton polygon of M = D q , K / D q , K P depends only on the formal isomorphism class of M. Caution ! By vectorialisation, equation L � system A � q -difference module M . By the cyclic vector lemma M = D q , K / D q , K P , where P is“dual”to L : they have symetric Newton polygons and opposite slopes.

An analogue of Generalities Stokes phenomenon for Fundamental solutions, constants q -difference equations One can prove that an analytic system σ q X = AX , Jacques Sauloy A ∈ GL n ( K ) always has a fundamental solution: Generalities X ∈ GL n ( M ( C ∗ , 0)) , Slopes Classification i.e. uniform in a punctured neighborhood of 0. Therefore, all uniform meromorphic solutions of σ q X = AX have the form X = X C , where C ∈ ( M ( C ∗ , 0) σ q ) n . The field of constants : M ( C ∗ , 0) σ q := { f ∈ M ( C ∗ , 0) | σ q f = f } can be identified with the field of elliptic functions M ( E q ), E q := C ∗ / q Z ≃ C / ( Z + Z τ ) , where e 2i πτ = q . (Identification through the map x �→ z := e 2i π x .)

An analogue of Generalities Stokes phenomenon for Associated vector bundle q -difference equations Jacques Sauloy This is for analytic systems (over K ). One defines: Generalities Slopes ( C ∗ , 0) × C n → ( C ∗ , 0) F (0) := ( z , X ) ∼ ( qz , A ( z ) X ) − z ∼ qz = E q . Classification A This is a holomorphic vector bundle over the complex torus (or elliptic curve) E q . The sheaf of holomorphic solutions of σ q X = AX near 0 is canonically isomorphic to the sheaf of sections of F (0) A A � F (0) is a“good”functor for classification and for Galois A theory (faithful, exact, ⊗ -compatible).

An analogue of Plan Stokes phenomenon for q -difference equations Jacques Sauloy Generalities Slopes Generalities on q -difference equations, systems and modules Classification The slope filtration Local analytic classification of irregular equations

An analogue of Slope filtration Stokes phenomenon for Pure modules, equations, systems q -difference equations Jacques Sauloy Generalities A module with one slope only is called pure isoclinic . Slopes Classification Pure isoclinic modules of slope 0 are fuchsian modules. They have the shape ( K n , Φ A ), with A ∈ GL n ( C ). Their analytic and formal classification (due to Birkhoff) are the same. Pure isoclinic modules of slope µ ∈ Z have the shape ( K n , Φ z µ A ), with A ∈ GL n ( C ). Their classification boils down to the fuchsian case. Pure isoclinic modules of nonintegral slope have been classified by van der Put and Reversat in 2005.

An analogue of Slope filtration Stokes phenomenon for The canonical filtration q -difference equations Theorem Jacques Sauloy Any q-difference module over K admits a unique filtration Generalities ( M ≤ µ ) µ ∈ Q such that each M ( µ ) := M ≤ µ M <µ is pure isoclinic of slope µ . The filtration is functorial and gr : M � � M ( µ ) is Slopes Classification a faithful exact C -linear ⊗ -compatible functor. Theorem Over ˆ K, the filtration splits canonically. After formalization (base change ˆ K ⊗ K − ), gr becomes isomorphic to the identity functor. Note that, contrary to the second, the first theorem has no equivalent in the case of differential equations: it is a consequence of Adams lemma (existence of an analytic factorisation for q -difference operators).

An analogue of Slope filtration Stokes phenomenon for Classification and graduation q -difference equations Jacques Sauloy A direct sum of pure isoclinic modules is called pure . Generalities Slopes Corollary Classification For pure modules, formal and analytic classification are equiv- alent. Formal classification of an analytic q-difference module M amounts to classification (formal or analytic) of the pure module grM. We already know: We want to study: The formal classification, The analytic classification i.e. classification of pure within a formal class, i.e. q -difference modules. with gr M fixed.

An analogue of Plan Stokes phenomenon for q -difference equations Jacques Sauloy Generalities Slopes Generalities on q -difference equations, systems and modules Classification The slope filtration Local analytic classification of irregular equations