Tutorial on Differential Galois Theory III

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1 Tutorial on Differential Galois Theory III T. Dyckerhoff Department of Mathematics University of Pennsylvania 02/14/08 / Oberflockenbach

2 Outline Today s plan Monodromy and singularities Riemann-Hilbert correspondence and applications Irregular singularities: Stokes approach Irregular singularities: Tannaka s approach Tannakian categories

3 Monodromy I Consider the -equation ( d y1 dz y 2 ) = ( 0 1 z 0 0 )( y1 y 2 ) with Y = ( 1 ln(z) 0 1 ) as fundamental solution matrix. ln(z) is not defined globally on C Analytic continuation: ln(z) ln(z) + 2πi

4 Monodromy I Consider the -equation ( d y1 dz y 2 ) = ( 0 1 z 0 0 )( y1 y 2 ) with Y = ( 1 ln(z) 0 1 ) as fundamental solution matrix. ln(z) is not defined globally on C Analytic continuation: ln(z) ln(z) + 2πi In our example ( ) 1 ln(z) Y = 0 1 ( 1 ln(z) 0 1 )( 1 2πi 0 1 )

5 Monodromy I Consider the -equation ( d y1 dz y 2 ) = ( 0 1 z 0 0 )( y1 y 2 ) with Y = ( 1 ln(z) 0 1 ) as fundamental solution matrix. ln(z) is not defined globally on C Analytic continuation: ln(z) ln(z) + 2πi In our example ( ) 1 ln(z) Y = 0 1 ( 1 ln(z) 0 1 )( 1 2πi 0 1 ) We obtain a representation π 1 (C ) GL 2 (C), n ( 1 2πi n 0 1 )

6 Monodromy II Given Equation d dz y = Ay with A C(z)n n The poles of A lie in a finite subset S P 1

7 Monodromy II Given Then Equation d dz y = Ay with A C(z)n n The poles of A lie in a finite subset S P 1 1 Solve [A] at a regular point p P 1 Y p GL n (O p )

8 Monodromy II Given Then Equation d dz y = Ay with A C(z)n n The poles of A lie in a finite subset S P 1 1 Solve [A] at a regular point p P 1 Y p GL n (O p ) 2 Continue Y p along a loop γ in P 1 \S based at p.

9 Monodromy II Given Then Equation d dz y = Ay with A C(z)n n The poles of A lie in a finite subset S P 1 1 Solve [A] at a regular point p P 1 Y p GL n (O p ) 2 Continue Y p along a loop γ in P 1 \S based at p. 3 Compare γy p = Y p C γ

10 Monodromy II Given Then Equation d dz y = Ay with A C(z)n n The poles of A lie in a finite subset S P 1 1 Solve [A] at a regular point p P 1 Y p GL n (O p ) 2 Continue Y p along a loop γ in P 1 \S based at p. 3 Compare Definition The map γy p = Y p C γ M : π 1 (P 1 \S, p) GL n (C), γ C γ is called the monodromy representation.

11 Riemann-Hilbert problem We obtain linear -equations with singularities in S P 1 M complex linear representations of π 1 (P 1 \S)

12 Riemann-Hilbert problem We obtain linear -equations with singularities in S P 1 M? Riemann-Hilbert problem complex linear representations of π 1 (P 1 \S)

13 Riemann-Hilbert problem We obtain linear -equations with singularities in S P 1 M? Riemann-Hilbert problem complex linear representations of π 1 (P 1 \S) This can t work: We considered the error function as a solution to the equation y + 2zy = 0. The equation is non-trivial (its solutions are non-elementary) Only one potential singularity at trivial monodromy

14 Analyzing the problem Look at y + 2zy = 0 around z = 1 put w = z 1 then d dz = w 2 d dw

15 Analyzing the problem Look at y + 2zy = 0 around z = 1 put w = z 1 then d dz = w 2 d 2 transformed equation: d equivalent system d dw ( y1 y 2 ) = dw dw 2 (y) ( 2 2 w 3 w ) d dw (y) = 0 with ( ( 2 w 3 2 w ) )( y1 The matrix defining the system has poles of order 3. y 2 )

16 Analyzing the problem Look at y + 2zy = 0 around z = 1 put w = z 1 then d dz = w 2 d 2 transformed equation: d equivalent system d dw ( y1 y 2 ) = dw dw 2 (y) ( 2 2 w 3 w ) d dw (y) = 0 with ( ( 2 w 3 2 w ) )( y1 The matrix defining the system has poles of order 3. Definition Two systems [A] and [B] are called gauge-equivalent over F C GL n (F) : C 1 AC C 1 (C) y 2 )

17 Analyzing the problem Look at y + 2zy = 0 around z = 1 put w = z 1 then d dz = w 2 d 2 transformed equation: d equivalent system d dw ( y1 y 2 ) = dw dw 2 (y) ( 2 2 w 3 w ) d dw (y) = 0 with ( ( 2 w 3 2 w ) )( y1 The matrix defining the system has poles of order 3. Definition Two systems [A] and [B] are called gauge-equivalent over F C GL n (F) : C 1 AC C 1 (C) A system [A] is called regular singular at x P 1 if it is gauge-equivalent over C({z x}) to a system with at most simple poles. Otherwise we say [A] is irregular singular at x. y 2 )

18 Riemann-Hilbert correspondence (2nd attempt) Theorem The functor linear -equations with regular singularities in S P 1 is an equivalence of categories. M complex linear representations of π 1 (P 1 \S)

19 Implications for differential Galois theory Given a A C(z) n n, the group M(π 1 (P 1 \S)) GL n is called the monodromy group. Theorem Assume A C(z) n n has only regular singularities. Then the differential Galois group of [A] is isomorphic to the Zariski closure of the monodromy group.

20 Implications for differential Galois theory Given a A C(z) n n, the group M(π 1 (P 1 \S)) GL n is called the monodromy group. Theorem Assume A C(z) n n has only regular singularities. Then the differential Galois group of [A] is isomorphic to the Zariski closure of the monodromy group. Proof. Let p P 1 be a regular (non-singular) point of A. 1 There exists a fundamental solution matrix Y p GL n (O p ).

21 Implications for differential Galois theory Given a A C(z) n n, the group M(π 1 (P 1 \S)) GL n is called the monodromy group. Theorem Assume A C(z) n n has only regular singularities. Then the differential Galois group of [A] is isomorphic to the Zariski closure of the monodromy group. Proof. Let p P 1 be a regular (non-singular) point of A. 1 There exists a fundamental solution matrix Y p GL n (O p ). 2 E = C(z)(Y p ) O p is a Picard-Vessiot field over C(z).

22 Implications for differential Galois theory Given a A C(z) n n, the group M(π 1 (P 1 \S)) GL n is called the monodromy group. Theorem Assume A C(z) n n has only regular singularities. Then the differential Galois group of [A] is isomorphic to the Zariski closure of the monodromy group. Proof. Let p P 1 be a regular (non-singular) point of A. 1 There exists a fundamental solution matrix Y p GL n (O p ). 2 E = C(z)(Y p ) O p is a Picard-Vessiot field over C(z). 3 If f E is invariant under monodromy then it extends to a holomorphic function on P 1 \S.

23 Implications for differential Galois theory Given a A C(z) n n, the group M(π 1 (P 1 \S)) GL n is called the monodromy group. Theorem Assume A C(z) n n has only regular singularities. Then the differential Galois group of [A] is isomorphic to the Zariski closure of the monodromy group. Proof. Let p P 1 be a regular (non-singular) point of A. 1 There exists a fundamental solution matrix Y p GL n (O p ). 2 E = C(z)(Y p ) O p is a Picard-Vessiot field over C(z). 3 If f E is invariant under monodromy then it extends to a holomorphic function on P 1 \S. 4 Use the fact that all solutions near a regular singular point have moderate growth O( z N ) to conclude that f is meromorphic.

24 Inverse problem over C(z) Given a linear algebraic group G over C, does there exist a -equation over C(z) which realizes G as its -Galois group?

25 Inverse problem over C(z) Given a linear algebraic group G over C, does there exist a -equation over C(z) which realizes G as its -Galois group? Theorem Yes.

26 Inverse problem over C(z) Given a linear algebraic group G over C, does there exist a -equation over C(z) which realizes G as its -Galois group? Theorem Yes. Lemma Every linear algebraic group G contains finitely many elements which generate G in the Zariski topology. Proof. By induction on dim(g).

27 Inverse problem over C(z) Given a linear algebraic group G over C, does there exist a -equation over C(z) which realizes G as its -Galois group? Theorem Yes. Proof. 1 Assume G is generated by C 1,..., C r GL n (C).

28 Inverse problem over C(z) Given a linear algebraic group G over C, does there exist a -equation over C(z) which realizes G as its -Galois group? Theorem Yes. Proof. 1 Assume G is generated by C 1,..., C r GL n (C). 2 Pick S = {s 1,..., s r, } P 1.

29 Inverse problem over C(z) Given a linear algebraic group G over C, does there exist a -equation over C(z) which realizes G as its -Galois group? Theorem Yes. Proof. 1 Assume G is generated by C 1,..., C r GL n (C). 2 Pick S = {s 1,..., s r, } P 1. 3 The group π 1 (P 1 \S) is freely generated by loops γ i around the points s i.

30 Inverse problem over C(z) Given a linear algebraic group G over C, does there exist a -equation over C(z) which realizes G as its -Galois group? Theorem Yes. Proof. 1 Assume G is generated by C 1,..., C r GL n (C). 2 Pick S = {s 1,..., s r, } P 1. 3 The group π 1 (P 1 \S) is freely generated by loops γ i around the points s i. 4 Choose the unique representation of π 1 (P 1 \S) which maps γ i to C i.

31 Inverse problem over C(z) Given a linear algebraic group G over C, does there exist a -equation over C(z) which realizes G as its -Galois group? Theorem Yes. Proof. 1 Assume G is generated by C 1,..., C r GL n (C). 2 Pick S = {s 1,..., s r, } P 1. 3 The group π 1 (P 1 \S) is freely generated by loops γ i around the points s i. 4 Choose the unique representation of π 1 (P 1 \S) which maps γ i to C i. 5 Apply Riemann-Hilbert.

32 Irregular singularities? - Stokes approach Riemann-Hilbert classifies regular singular equations by monodromy data. Classification of irregular singularities Recall: regularity of a singularity moderate growth of solutions

33 Irregular singularities? - Stokes approach Riemann-Hilbert classifies regular singular equations by monodromy data. Classification of irregular singularities Recall: regularity of a singularity moderate growth of solutions A solution near an irregular singularity can have a wild growth behaviour which depends on the direction θ from which we approach the singular point (e.g. exp( 1 z 2 ))

34 Irregular singularities? - Stokes approach Riemann-Hilbert classifies regular singular equations by monodromy data. Classification of irregular singularities Recall: regularity of a singularity moderate growth of solutions A solution near an irregular singularity can have a wild growth behaviour which depends on the direction θ from which we approach the singular point (e.g. exp( 1 z 2 )) Let V θ be the space of solutions on a small sector around θ

35 Irregular singularities? - Stokes approach Riemann-Hilbert classifies regular singular equations by monodromy data. Classification of irregular singularities Recall: regularity of a singularity moderate growth of solutions A solution near an irregular singularity can have a wild growth behaviour which depends on the direction θ from which we approach the singular point (e.g. exp( 1 z 2 )) Let V θ be the space of solutions on a small sector around θ We can decompose V θ into a direct sum of subspaces defined by the various growth rates

36 Irregular singularities? - Stokes approach Riemann-Hilbert classifies regular singular equations by monodromy data. Classification of irregular singularities Recall: regularity of a singularity moderate growth of solutions A solution near an irregular singularity can have a wild growth behaviour which depends on the direction θ from which we approach the singular point (e.g. exp( 1 z 2 )) Let V θ be the space of solutions on a small sector around θ We can decompose V θ into a direct sum of subspaces defined by the various growth rates The obstruction for the existence of a global growth decomposition around the singularity is called the Stokes phenomenon

37 Irregular singularities? - Stokes approach Theorem (Deligne,Turittin,Hukuhara,Malgrange) There is a functor linear -equations with arbitrary singularities in S P 1 which is an equivalence. = representations of π 1 (P 1 \S) + Stokes data for S

38 Irregular singularities? - Stokes approach Theorem (Deligne,Turittin,Hukuhara,Malgrange) There is a functor linear -equations with arbitrary singularities in S P 1 which is an equivalence. = representations of π 1 (P 1 \S) + Stokes data for S Again we can use the classification data to compute the differential Galois group, at least locally. Theorem (Ramis) The differential Galois group of an equation over the field of convergent Laurent series C({z}) is topologically generated by the monodromy, the Stokes matrices and the so-called Ramis torus.

39 Irregular singularities? - Tannaka s approach The failure of Riemann-Hilbert for irregular singularities can be regarded as a motivation for the Tannakian approach. linear -equations with regular singularities in S P 1 = complex linear representations of π top 1 (P1 \S) linear -equations with arbitrary singularities in S P 1 =?

40 Irregular singularities? - Tannaka s approach The failure of Riemann-Hilbert for irregular singularities can be regarded as a motivation for the Tannakian approach. linear -equations with regular singularities in S P 1 linear -equations with arbitrary singularities in S P 1 = = complex linear representations of π top 1 (P1 \S) representations of a suitable group π diff 1 (P1 \S) In this sense, the group π1 diff can be understood as an enhancement of the topological fundamental group π top 1. It is called the Tannakian fundamental group.

41 Tannakian categories Definition A Tannakian category over K is a rigid abelian tensor category C over K with an exact faithful K -linear functor called the fiber functor. ω : C (Vect/K)

42 Tannakian categories Definition A Tannakian category over K is a rigid abelian tensor category C over K with an exact faithful K -linear functor called the fiber functor. ω : C (Vect/K) Theorem (Deligne, Grothendieck, Milne, Saavedra) Let C be a Tannakian category over K. Then Aut (ω) is isomorphic to an affine group scheme G over K. there is an equivalence of abelian tensor categories C ω = (Rep(G)/K)

43 Examples of Tannakian categories (I) Z-graded vector spaces Let V complex representation of the group G m, C V admits a weight space decomposition V = char(g m ) = Z χ char(g m) V χ

44 Examples of Tannakian categories (I) Z-graded vector spaces Let V complex representation of the group G m, C V admits a weight space decomposition V = char(g m ) = Z χ char(g m) The category of Z-graded complex vector spaces is Tannakian over C with fundamental group G m. V χ

45 Examples of Tannakian categories (II) Real Hodge structures Consider the group G = C /R as a real algebraic group

46 Examples of Tannakian categories (II) Real Hodge structures Consider the group G = C /R as a real algebraic group C R G = G m G m char(c R G) = Z Z

47 Examples of Tannakian categories (II) Real Hodge structures Consider the group G = C /R as a real algebraic group C R G = G m G m char(c R G) = Z Z Let V be a real representation of G V R C = (p,q) Z Z V p,q V p,q = V q,p

48 Examples of Tannakian categories (II) Real Hodge structures Consider the group G = C /R as a real algebraic group C R G = G m G m char(c R G) = Z Z Let V be a real representation of G V R C = (p,q) Z Z V p,q V p,q = V q,p The category of real Hodge structures is Tannakian over R with fundamental group C /R.

49 D-modules Definition Let F be a -field. Define the non-commutative ring D = F[ ] subject to f = f + (f) of differential operators over F. A D-module is module over D, finite dimensional over F.

50 D-modules vs. -equations Let M be a D-module. Let e = (e 1,..., e n ) be an F -basis. There exists A F n n such (e) = ea

51 D-modules vs. -equations Let M be a D-module. Let e = (e 1,..., e n ) be an F -basis. There exists A F n n such (e) = ea Assume f = (f 1,..., f n ) is another basis with (f) = f B then there exists C GL n (F) such that e = f C

52 D-modules vs. -equations Let M be a D-module. Let e = (e 1,..., e n ) be an F -basis. There exists A F n n such (e) = ea Assume f = (f 1,..., f n ) is another basis with (f) = f B then there exists C GL n (F) such that e = f C So ea = (e) = (f C) = (f)c + f (C) = e(c 1 BC + C 1 (C))

53 D-modules vs. -equations Let M be a D-module. Let e = (e 1,..., e n ) be an F -basis. There exists A F n n such (e) = ea Assume f = (f 1,..., f n ) is another basis with (f) = f B then there exists C GL n (F) such that e = f C So ea = (e) = (f C) = (f)c + f (C) = e(c 1 BC + C 1 (C)) A = C 1 BC + C 1 (C) Conclusion A D-module is an intrinsic description of a gauge-equivalence class of -equations.

54 The ultimate example of a Tannakian category Theorem Let F -field with algebraically closed field of constants. Then the category of D-modules is a Tannakian category.

55 The ultimate example of a Tannakian category Theorem Let F -field with algebraically closed field of constants. Then the category of D-modules is a Tannakian category. Example Let M be a D-module. The full tensor subcategory M (D-mod) generated by M is a Tannakian category.

56 The ultimate example of a Tannakian category Theorem Let F -field with algebraically closed field of constants. Then the category of D-modules is a Tannakian category. Example Let M be a D-module. The full tensor subcategory M (D-mod) generated by M is a Tannakian category. Theorem The Tannakian fundamental group of M is isomorphic to the differential Galois group of any equation in the gauge equivalence class given by M.

57 A reality check with Tannaka Theorem Assume A C(z) n n with trace(a) = 0. Then the -Galois group G of A is isomorphic to a subgroup of SL n (C). Proof. Let M be the D-module associated to A. 1 Applying the fiber functor to M yields an n-dimensional representation V of the Galois group G (the usual standard representation on the solution space of [A])

58 A reality check with Tannaka Theorem Assume A C(z) n n with trace(a) = 0. Then the -Galois group G of A is isomorphic to a subgroup of SL n (C). Proof. Let M be the D-module associated to A. 1 Applying the fiber functor to M yields an n-dimensional representation V of the Galois group G (the usual standard representation on the solution space of [A]) 2 Tannakian duality: all constructions involving M must be compatible with constructions involving V

59 A reality check with Tannaka Theorem Assume A C(z) n n with trace(a) = 0. Then the -Galois group G of A is isomorphic to a subgroup of SL n (C). Proof. Let M be the D-module associated to A. 1 Applying the fiber functor to M yields an n-dimensional representation V of the Galois group G (the usual standard representation on the solution space of [A]) 2 Tannakian duality: all constructions involving M must be compatible with constructions involving V 3 check: the D-module associated to the 1 1-matrix trace(a) is exactly n M

60 A reality check with Tannaka Theorem Assume A C(z) n n with trace(a) = 0. Then the -Galois group G of A is isomorphic to a subgroup of SL n (C). Proof. Let M be the D-module associated to A. 1 Applying the fiber functor to M yields an n-dimensional representation V of the Galois group G (the usual standard representation on the solution space of [A]) 2 Tannakian duality: all constructions involving M must be compatible with constructions involving V 3 check: the D-module associated to the 1 1-matrix trace(a) is exactly n M 4 trace(a) = 0 implies n M is trivial

61 A reality check with Tannaka Theorem Assume A C(z) n n with trace(a) = 0. Then the -Galois group G of A is isomorphic to a subgroup of SL n (C). Proof. Let M be the D-module associated to A. 1 Applying the fiber functor to M yields an n-dimensional representation V of the Galois group G (the usual standard representation on the solution space of [A]) 2 Tannakian duality: all constructions involving M must be compatible with constructions involving V 3 check: the D-module associated to the 1 1-matrix trace(a) is exactly n M 4 trace(a) = 0 implies n M is trivial 5 Tannaka: n V is trivial

62 A reality check with Tannaka Theorem Assume A C(z) n n with trace(a) = 0. Then the -Galois group G of A is isomorphic to a subgroup of SL n (C). Proof. Let M be the D-module associated to A. 1 Applying the fiber functor to M yields an n-dimensional representation V of the Galois group G (the usual standard representation on the solution space of [A]) 2 Tannakian duality: all constructions involving M must be compatible with constructions involving V 3 check: the D-module associated to the 1 1-matrix trace(a) is exactly n M 4 trace(a) = 0 implies n M is trivial 5 Tannaka: n V is trivial 6 know: n V is the determinant representation of V

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