Linear ODEs and ruled surfaces

Transcription

1 Linear ODEs and ruled surfaces Camilo Sanabria CUNY Bronx Community College de los Andes University, Bogotá January 6 th, 2012

2 Klein s Theorem An early result of F. Klein states that: If y + a(x)y + b(x)y = 0 (a(x), b(x) C(x)) has finite projective Galois group G,

3 Klein s Theorem An early result of F. Klein states that: If y + a(x)y + b(x)y = 0 (a(x), b(x) C(x)) has finite projective Galois group G, then the solutions to the equation are of the form: f (x) 2F 1(α,β,γ) (P(x))

4 Klein s Theorem An early result of F. Klein states that: If y + a(x)y + b(x)y = 0 (a(x), b(x) C(x)) has finite projective Galois group G, then the solutions to the equation are of the form: f (x) 2F 1(α,β,γ) (P(x)) where f (x) is a solution to a first order homogeneous linear differential equation

5 Klein s Theorem An early result of F. Klein states that: If y + a(x)y + b(x)y = 0 (a(x), b(x) C(x)) has finite projective Galois group G, then the solutions to the equation are of the form: f (x) 2F 1(α,β,γ) (P(x)) where f (x) is a solution to a first order homogeneous linear differential equation, 2 F 1(α,β,γ) (x) is a hypergeometric function

6 Klein s Theorem An early result of F. Klein states that: If y + a(x)y + b(x)y = 0 (a(x), b(x) C(x)) has finite projective Galois group G, then the solutions to the equation are of the form: f (x) 2F 1(α,β,γ) (P(x)) where f (x) is a solution to a first order homogeneous linear differential equation, 2 F 1(α,β,γ) (x) is a hypergeometric function and P(x) C(x).

7 Klein s Theorem f (x) 2F 1(α,β,γ) (P(x)) In modern terms we say y + a(x)y + b(x)y = 0 is projectively equivalent to the pullback of a hypergeometric equation by a rational map.

8 Klein s Theorem f (x) 2F 1(α,β,γ) (P(x)) In modern terms we say y + a(x)y + b(x)y = 0 is projectively equivalent to the pullback of a hypergeometric equation by a rational map. The result was revisited by Dwork and Baldassari and algorithmically implemented in a joint work of M. Berkenbosch, M. van Hoeij and J.A. Weil.

9 Klein s Theorem f (x) 2F 1(α,β,γ) (P(x)) In modern terms we say y + a(x)y + b(x)y = 0 is projectively equivalent to the pullback of a hypergeometric equation by a rational map. The result was revisited by Dwork and Baldassari and algorithmically implemented in a joint work of M. Berkenbosch, M. van Hoeij and J.A. Weil. The three latter authors realized that the f (x) and the P(x) can be computed very efficiently.

10 Klein s Theorem f (x) 2F 1(α,β,γ) (P(x)) In modern terms we say y + a(x)y + b(x)y = 0 is projectively equivalent to the pullback of a hypergeometric equation by a rational map. The result was revisited by Dwork and Baldassari and algorithmically implemented in a joint work of M. Berkenbosch, M. van Hoeij and J.A. Weil. The three latter authors realized that the f (x) and the P(x) can be computed very efficiently. The triples (α, β, γ) correspond to the family of Galois coverings of the Riemann sphere by the Riemann sphere.

11 Klein s Theorem f (x) 2F 1(α,β,γ) (P(x)) In modern terms we say y + a(x)y + b(x)y = 0 is projectively equivalent to the pullback of a hypergeometric equation by a rational map. The result was revisited by Dwork and Baldassari and algorithmically implemented in a joint work of M. Berkenbosch, M. van Hoeij and J.A. Weil. The three latter authors realized that the f (x) and the P(x) can be computed very efficiently. The triples (α, β, γ) correspond to the family of Galois coverings of the Riemann sphere by the Riemann sphere. So, one triple for each finite group G PSL 2 (C).

12 Klein s Theorem The argument behind Klein s result goes as follows: two C-linearly independent solutions of the equation y 0, y 1 : U P 1 (C) C

13 Klein s Theorem The argument behind Klein s result goes as follows: two C-linearly independent solutions of the equation define an analytic map y 0, y 1 : U P 1 (C) C Φ : U P 1 (C) P 1 (C) x (y 0 (x) : y 1 (x))

14 Klein s Theorem P 1 (C) Φ U

15 Klein s Theorem P 1 (C) Φ U letting the projective Galois group G act by Mobiüs transformations on P 1 (C), we can postcompose Φ by the quotient P 1 (C) P 1 (C)/G

16 Klein s Theorem P 1 (C) Φ U letting the projective Galois group G act by Mobiüs transformations on P 1 (C), we can postcompose Φ by the quotient P 1 (C) P 1 (C)/G P 1 (C)

17 Klein s Theorem U Φ Φ G P 1 (C) P 1 (C)/G

18 Klein s Theorem U Φ Φ G P 1 (C) P 1 (C)/G by Galois correspondence the bottom arrow is algebraic;

19 Klein s Theorem U Φ Φ G P 1 (C) P 1 (C)/G by Galois correspondence the bottom arrow is algebraic; and therefore Φ G can be extended to an algebraic map Φ G : P 1 (C) P 1 (C) P 1 (C)/G

20 Klein s Theorem Φ G : P 1 (C) P 1 (C)/G x (y 0 (x) : y 1 (x)) G If Φ G is an isomorphism, then the equation is projectively equivalent to a hypergeometric equation.

21 Klein s Theorem Φ G : P 1 (C) P 1 (C)/G x (y 0 (x) : y 1 (x)) G If Φ G is an isomorphism, then the equation is projectively equivalent to a hypergeometric equation. If not, then it is projectively equivalent to the pullback, by Φ G, of a hypergeometric equation.

22 Standard Equations f (x) 2F 1(α,β,γ) (P(x)) The collection of hypergeometric equations corresponding to the triples (α, β, γ) were named by the three authors as Standard Equations. M. Berkenbosch extended Klein s result by broadening the concept of standard equation to third order.

23 Standard Equations One can actually easily extend the concept to equations of any order as follows: P n 1 (C) Φ U P n 1 (C)/G Φ G

24 Standard Equations One can actually easily extend the concept to equations of any order as follows: where P n 1 (C) Φ U P n 1 (C)/G Φ G Φ : U P 1 (C) P n 1 (C) x (y 0 (x) : y 1 (x) :... : y n 1 (x)) and the y i s are C-linearly independent solutions.

25 Standard Equations One can actually easily extend the concept to equations of any order as follows: where P n 1 (C) Φ U P n 1 (C)/G Φ G Φ : U P 1 (C) P n 1 (C) x (y 0 (x) : y 1 (x) :... : y n 1 (x)) and the y i s are C-linearly independent solutions. Again, the equation is standard if Φ G is an embedding;

26 Standard Equations One can actually easily extend the concept to equations of any order as follows: where P n 1 (C) Φ U P n 1 (C)/G Φ G Φ : U P 1 (C) P n 1 (C) x (y 0 (x) : y 1 (x) :... : y n 1 (x)) and the y i s are C-linearly independent solutions. Again, the equation is standard if Φ G is an embedding; if not, one can prove that, the equation is the pullback of one.

27 Standard Equations & Ruled Surfaces The issue now is that the collection of standard equations, for orders bigger than 2, cannot be classified by a structured family of coverings.

28 Standard Equations & Ruled Surfaces The issue now is that the collection of standard equations, for orders bigger than 2, cannot be classified by a structured family of coverings. Moreover, for a given group G PGL n (C) there are infinitely many standard equations.

29 Standard Equations & Ruled Surfaces The issue now is that the collection of standard equations, for orders bigger than 2, cannot be classified by a structured family of coverings. Moreover, for a given group G PGL n (C) there are infinitely many standard equations. We will classify the standard equations instead using ruled surfaces:

30 Standard Equations & Ruled Surfaces The issue now is that the collection of standard equations, for orders bigger than 2, cannot be classified by a structured family of coverings. Moreover, for a given group G PGL n (C) there are infinitely many standard equations. We will classify the standard equations instead using ruled surfaces: A ruled surface S is the total space of a bundle over a projective curve C where every fiber is isomorphic to P 1 (C): π : S C with π 1 (x) P 1 (C)

31 Ruled Surfaces The simplest ruled surface is P 1 (C) P 1 (C).

32 Ruled Surfaces The simplest ruled surface is P 1 (C) P 1 (C). In general, a ruled surface over C corresponds to the bundle obtained by the projective spaces defined by the fibers of a rank-2 vector bundle over C: S = P(O C (n) O C (m)).

33 Ruled Surfaces The simplest ruled surface is P 1 (C) P 1 (C). In general, a ruled surface over C corresponds to the bundle obtained by the projective spaces defined by the fibers of a rank-2 vector bundle over C: S = P(O C (n) O C (m)). The isomorphism class of the ruled surface S is determined by the quantity m n.

34 Linear ODEs & Ruled Surfaces We get our ruled surface as follows: U P n 1 (C) P n 1 (C) Ψ P n 1 (C)/G P n 1 (C)/G Ψ G

35 Linear ODEs & Ruled Surfaces We get our ruled surface as follows: U P n 1 (C) P n 1 (C) Ψ P n 1 (C)/G P n 1 (C)/G Ψ G where Ψ : U P 1 (C) P n 1 (C) P n 1 (C) x (y 0 (x) : y 1 (x) :... : y n 1 (x), y 0(x) : y 1(x) :... : y n 1(x)) (the y i s are C-linearly independent solutions).

36 Linear ODEs & Ruled Surfaces Ψ G : P 1 (C) P n 1 (C)/G P n 1 (C)/G x (y 0 (x) : y 1 (x) :... : y n 1 (x), y 0(x) : y 1(x) :... : y n 1(x)) G

37 Linear ODEs & Ruled Surfaces Ψ G : P 1 (C) P n 1 (C)/G P n 1 (C)/G x (y 0 (x) : y 1 (x) :... : y n 1 (x), y 0(x) : y 1(x) :... : y n 1(x)) G The result is: Two linear ODEs with the same projective Galois group G are projectively equivalent to the pullback of the same standard equation if the ruled surfaces defined by their corresponding Ψ G coincide.

38 Examples ( ) A 4 P O(2) O(26)

39 Examples ( ) A 4 P O(2) O(26) ( ) S 4 P O(1) O(25)

40 Examples ( ) A 4 P O(2) O(26) ( ) S 4 P O(1) O(25) ( ) A 5 P O(1) O(61)

41 Examples ( ) A 4 P O(2) O(26) ( ) S 4 P O(1) O(25) ( ) A 5 P O(1) O(61) ( ) D 2 n P O(2) O(2[2n + 1]) if 2 n

42 Examples ( ) A 4 P O(2) O(26) ( ) S 4 P O(1) O(25) ( ) A 5 P O(1) O(61) ( ) D 2 n P O(2) O(2[2n + 1]) if 2 n ( ) D 2 n P O(1) O(2n + 1) if 2 n

43 References F. Baldassarri, B. Dwork, On second order linear differential equations with algebraic solutions, Amer. J. Math. 101 (1979), no. 1, M. Berkenbosch, Algorithms and moduli spaces for differential equations, Séminaires & Congrès 13 (2006) 1-38.