Inverse Galois Problem for C(t)

Transcription

1 Inverse Galois Problem for C(t) Padmavathi Srinivasan PuMaGraSS March 2, 2012

2 Outline 1 The problem 2 Compact Riemann Surfaces 3 Covering Spaces 4 Connection to field theory 5 Proof of the Main theorem 6 References

3 Question Does every finite group occur as the Galois group of some finite Galois extension L of Q? Statement

4 Statement Question Does every finite group occur as the Galois group of some finite Galois extension L of Q? This is still an open problem!

5 Statement Question Does every finite group occur as the Galois group of some finite Galois extension L of Q? This is still an open problem! Question Does every finite group occur as the Galois group of some finite Galois extension L of C(t)?

6 Statement Question Does every finite group occur as the Galois group of some finite Galois extension L of Q? This is still an open problem! Question Does every finite group occur as the Galois group of some finite Galois extension L of C(t)? Yes!

7 Outline 1 The problem 2 Compact Riemann Surfaces 3 Covering Spaces 4 Connection to field theory 5 Proof of the Main theorem 6 References

8 Definition A Riemann surface is a one dimensional complex manifold.

9 Definition A Riemann surface is a one dimensional complex manifold. Examples: C, P 1 (C), C/Γ for a complete lattice Γ C, Smooth affine plane curve in C 2

10 Definition A Riemann surface is a one dimensional complex manifold. Examples: C, P 1 (C), C/Γ for a complete lattice Γ C, Smooth affine plane curve in C 2 Definition Let X and Y be two Riemann surfaces. A holomorphic map φ : Y X is a continuous map such that for every pair of charts (V, g) of Y and (U, f ) of X such that φ(v ) U, the functions f φ g 1 : g(v ) C are holomorphic.

11 Definition A Riemann surface is a one dimensional complex manifold. Examples: C, P 1 (C), C/Γ for a complete lattice Γ C, Smooth affine plane curve in C 2 Definition Let X and Y be two Riemann surfaces. A holomorphic map φ : Y X is a continuous map such that for every pair of charts (V, g) of Y and (U, f ) of X such that φ(v ) U, the functions f φ g 1 : g(v ) C are holomorphic. Examples: The map z z k on C for all k 1, the natural projection map π : C C/Γ

12 Proposition Local structure of holomorphic maps Let φ : Y X be a non-constant holomorphic map of Riemann surfaces, and y a point of Y with image x = φ(y) in X. There exist charts (V y, g y ) of Y and (U x, f x ) of X such that V y is an open neighbourhood of y, U x an open neighbourhood of x satisfying φ(v y ) U x and g y (y) = f x (x) = 0 such that the diagram V y φ U x g y f x C z z ey commutes with an appropriate positive integer e y that does on depend on the choice of the complex charts. C

13 Corollaries Definition Let φ : Y X, e y etc. be defined as before. The integer e y is called the ramification index of φ at y. The points y with e y > 1 are called branch points. The set of branch points of φ will be denoted S φ.

14 Corollaries Definition Let φ : Y X, e y etc. be defined as before. The integer e y is called the ramification index of φ at y. The points y with e y > 1 are called branch points. The set of branch points of φ will be denoted S φ. Corollary A non-constant holomorphic map between Riemann surfaces is open.

15 Corollaries Definition Let φ : Y X, e y etc. be defined as before. The integer e y is called the ramification index of φ at y. The points y with e y > 1 are called branch points. The set of branch points of φ will be denoted S φ. Corollary A non-constant holomorphic map between Riemann surfaces is open. Corollary Let φ : Y X be a non-constant holomorphic map between two Riemann surfaces. The fibres of φ and the set S φ are discrete closed subsets of Y.

16 Corollaries Corollary Let φ : Y X be a non-constant holomorphic map between two connected Riemann surfaces. If Y is compact, then X is compact and φ is surjective.

17 Corollaries Corollary Let φ : Y X be a non-constant holomorphic map between two connected Riemann surfaces. If Y is compact, then X is compact and φ is surjective. Corollary Every holomorphic function on a compact Riemann surface (holomorphic map from the compact Riemann surface to C) is constant.

18 Corollaries Corollary Let φ : Y X be a non-constant holomorphic map between two connected Riemann surfaces. If Y is compact, then X is compact and φ is surjective. Corollary Every holomorphic function on a compact Riemann surface (holomorphic map from the compact Riemann surface to C) is constant. Corollary Every holomorphic map from P 1 to itself is rational i.e., can be written as the quotient of two polynomials.

19 Meromorphic functions Definition Let X be a Riemann surface. A meromorphic function φ on X is a holomorphic function on X \ S where S X is a discrete closed subset and further, for all charts (U, f ), the function φ f 1 : f (U) C is meromorphic.

20 Meromorphic functions Definition Let X be a Riemann surface. A meromorphic function φ on X is a holomorphic function on X \ S where S X is a discrete closed subset and further, for all charts (U, f ), the function φ f 1 : f (U) C is meromorphic. Meromorphic functions on X form a ring with respect to the usual addition and multiplication of functions. We denote this ring by M(X ).

21 Meromorphic functions Definition Let X be a Riemann surface. A meromorphic function φ on X is a holomorphic function on X \ S where S X is a discrete closed subset and further, for all charts (U, f ), the function φ f 1 : f (U) C is meromorphic. Meromorphic functions on X form a ring with respect to the usual addition and multiplication of functions. We denote this ring by M(X ). Also, it is easy to see that meromorphic functions on X are the same as holomorphic maps from X to P 1.

22 Meromorphic functions Definition Let X be a Riemann surface. A meromorphic function φ on X is a holomorphic function on X \ S where S X is a discrete closed subset and further, for all charts (U, f ), the function φ f 1 : f (U) C is meromorphic. Meromorphic functions on X form a ring with respect to the usual addition and multiplication of functions. We denote this ring by M(X ). Also, it is easy to see that meromorphic functions on X are the same as holomorphic maps from X to P 1. Lemma If X is connected, the ring M(X ) is a field.

23 Meromorphic functions The lemma we just stated is an easy consequence of the following theorem.

24 Meromorphic functions The lemma we just stated is an easy consequence of the following theorem. Theorem (Identity Theorem) Suppose X and Y Riemann surfaces, with X connected and f 1, f 2 : X Y are two holomorphic maps that coincide on a set A having a limit point a X. Then f 1 and f 2 are identically equal.

25 Meromorphic functions The lemma we just stated is an easy consequence of the following theorem. Theorem (Identity Theorem) Suppose X and Y Riemann surfaces, with X connected and f 1, f 2 : X Y are two holomorphic maps that coincide on a set A having a limit point a X. Then f 1 and f 2 are identically equal. What we proved before as a corollary to the proposition can be restated as saying M(P 1 ) = C(t).

26 Connection to topological covers Definition A continuous map of locally compact topological spaces (L.C.T.S.) is proper if the preimage of every compact set is compact.

27 Connection to topological covers Definition A continuous map of locally compact topological spaces (L.C.T.S.) is proper if the preimage of every compact set is compact. Examples: Any continuous map between L.C.T.S. where the domain is compact and range is Hausdorff, Any finite cover p : Y X of L.C.T.S.

28 Connection to topological covers Definition A continuous map of locally compact topological spaces (L.C.T.S.) is proper if the preimage of every compact set is compact. Examples: Any continuous map between L.C.T.S. where the domain is compact and range is Hausdorff, Any finite cover p : Y X of L.C.T.S. Theorem Let X be a connected Riemann surface, and φ : Y X a proper holomorphic map. The map φ is surjective with finite fibres, and its restriction to Y \ φ 1 (φ(s φ )) is a finite topological cover of X \ φ(s φ ).

29 Connection to topological covers Definition A continuous map of locally compact topological spaces (L.C.T.S.) is proper if the preimage of every compact set is compact. Examples: Any continuous map between L.C.T.S. where the domain is compact and range is Hausdorff, Any finite cover p : Y X of L.C.T.S. Theorem Let X be a connected Riemann surface, and φ : Y X a proper holomorphic map. The map φ is surjective with finite fibres, and its restriction to Y \ φ 1 (φ(s φ )) is a finite topological cover of X \ φ(s φ ). Such a map will be called a finite branched cover, and the degree of this map will be the degree of the topological cover obtained from it as above.

30 Connection to topological covers A proper holomorphic map is determined by its topological properties. It can formally be stated as an equivalence of catgeories; the category of Riemann surfaces with a map onto X, unbranched over X \ S is equivalent to the category of topological covers of X \ S.

31 Connection to topological covers A proper holomorphic map is determined by its topological properties. It can formally be stated as an equivalence of catgeories; the category of Riemann surfaces with a map onto X, unbranched over X \ S is equivalent to the category of topological covers of X \ S. Lemma Let X be a Riemann surface, Y a topological space with a covering map p : Y X. The space Y can be endowed with a unique complex structure for which p becomes a holomorphic mapping.

32 Connection to topological covers A proper holomorphic map is determined by its topological properties. It can formally be stated as an equivalence of catgeories; the category of Riemann surfaces with a map onto X, unbranched over X \ S is equivalent to the category of topological covers of X \ S. Lemma Let X be a Riemann surface, Y a topological space with a covering map p : Y X. The space Y can be endowed with a unique complex structure for which p becomes a holomorphic mapping. Lemma Given two Riemann surfaces Y and Z equipped with proper non-constant holomorphic maps φ Y and φ Z onto X with all branch points above S and a morphism of covers ρ : Y Z over X where X = X \ S, Y = Y \ φ 1 Y (S) and Z = Z \ φ 1 Z (S), there is a unique holomorphic map ρ : Y Z over X extending ρ.

33 Connection to topological covers Proposition Assume we are given a connected Riemann surface X, a discrete closed set S of points of X and a finite connected cover φ : Y X, where X = X \ S. Then there exists a Riemann surface Y containing Y as an open subset and a proper holomorphic map φ : Y X such that φ Y = φ and Y = Y \ φ 1 (S).

34 Connection to topological covers Proposition Assume we are given a connected Riemann surface X, a discrete closed set S of points of X and a finite connected cover φ : Y X, where X = X \ S. Then there exists a Riemann surface Y containing Y as an open subset and a proper holomorphic map φ : Y X such that φ Y = φ and Y = Y \ φ 1 (S). Corollary Given a map p : Y X as above, the group of automorphisms of the space Y over X, Aut (Y X ) as branched covers over X (i.e. compatible with p) is isomorphic to the group Aut (Y X ) where these are the automorphisms of Y as a topological cover over X.

35 Outline 1 The problem 2 Compact Riemann Surfaces 3 Covering Spaces 4 Connection to field theory 5 Proof of the Main theorem 6 References

36 Galois covers Given a cover p : Y X, its automorphisms are to be automorphisms of Y as a space over X i.e. automorphisms compatible with p. This is a group with respect to composition, and we will denote it Aut (Y X ). If we fix a point x X, the group Aut (Y X ) maps the set p 1 (x) onto itself, so p 1 (x) has a natural Aut (Y X ) action.

37 Galois covers Given a cover p : Y X, its automorphisms are to be automorphisms of Y as a space over X i.e. automorphisms compatible with p. This is a group with respect to composition, and we will denote it Aut (Y X ). If we fix a point x X, the group Aut (Y X ) maps the set p 1 (x) onto itself, so p 1 (x) has a natural Aut (Y X ) action. Definition A connected cover p : Y X is said to be Galois if Aut (Y X ) acts transitively on each fibre of p.

38 Galois covers Given a cover p : Y X, its automorphisms are to be automorphisms of Y as a space over X i.e. automorphisms compatible with p. This is a group with respect to composition, and we will denote it Aut (Y X ). If we fix a point x X, the group Aut (Y X ) maps the set p 1 (x) onto itself, so p 1 (x) has a natural Aut (Y X ) action. Definition A connected cover p : Y X is said to be Galois if Aut (Y X ) acts transitively on each fibre of p. Example: The exponential map exp : R S 1 is a Galois cover with Aut (R S 1 ) = Z.

39 Fundamental theorem of Galois covers Theorem Let p : Y X be a Galois cover. For each subgroup H of G = Aut (Y X ), the projection p induces a natural map p H : H \ Y X which turns H \ Y into a cover of X. Conversely, if Z X is a connected cover fitting into a commutative diagram Y f Z p then f : Y Z is a Galois cover and actually Z = H \ Y for the subgroup H = Aut (Y Z) of G. The maps H H \ Y, Z Aut(Y Z) induce a bijection between subgroups of G and intermediate covers Z as above. The cover q : Z X is Galois if and only if H is a normal subgroup of G, in which case Aut (Z X ) = G/H. X q

40 Universal cover Amongst all covering spaces of a manifold X, there is a largest one dubbed the universal cover and all other covering spaces can be obtained as quotients of this space. The automorphism group of this space can be determined by data intrinsic to X, namely the fundamental group of X.

41 Universal cover Amongst all covering spaces of a manifold X, there is a largest one dubbed the universal cover and all other covering spaces can be obtained as quotients of this space. The automorphism group of this space can be determined by data intrinsic to X, namely the fundamental group of X. Definition Suppose X and Y are connected topological spaces and p : Y X is a covering map. p : Y X is called the universal covering of X if it satisfies the following universal property. For every covering map q : Z X, with Z connected, and every choice of points y 0 Y, z 0 Z with p(y 0 ) = q(z 0 ), there exists exactly one continuous fiber-preserving map f : Y Z such that f (y 0 ) = z 0.

42 Universal cover Amongst all covering spaces of a manifold X, there is a largest one dubbed the universal cover and all other covering spaces can be obtained as quotients of this space. The automorphism group of this space can be determined by data intrinsic to X, namely the fundamental group of X. Definition Suppose X and Y are connected topological spaces and p : Y X is a covering map. p : Y X is called the universal covering of X if it satisfies the following universal property. For every covering map q : Z X, with Z connected, and every choice of points y 0 Y, z 0 Z with p(y 0 ) = q(z 0 ), there exists exactly one continuous fiber-preserving map f : Y Z such that f (y 0 ) = z 0. Such a cover is easily seen to be Galois, if it exists, and is unique up to unique isomorphism.

43 Theorem The universal cover exists for a connected and locally simply connected topological space X. Universal cover

44 Universal cover Theorem The universal cover exists for a connected and locally simply connected topological space X. Theorem Let X be a path-connected and locally simply connected space. A cover X X is universal if an only if X is simply connected.

45 Universal cover Theorem The universal cover exists for a connected and locally simply connected topological space X. Theorem Let X be a path-connected and locally simply connected space. A cover X X is universal if an only if X is simply connected. Theorem The universal cover X is a connected Galois cover of X, with automorphism group isomorphic to π 1 (X ).

46 Outline 1 The problem 2 Compact Riemann Surfaces 3 Covering Spaces 4 Connection to field theory 5 Proof of the Main theorem 6 References

47 Existence of enough meromorphic functions Theorem Let X be a compact Riemann surface, x 1,..., x n X a finite set of points, and a 1,..., a n a finite set of complex numbers. Then there exists a meromorphic function f M(X ) such that f is holomorphic at all the x i and f (x i ) = a i for 1 i n.

48 Existence of enough meromorphic functions Theorem Let X be a compact Riemann surface, x 1,..., x n X a finite set of points, and a 1,..., a n a finite set of complex numbers. Then there exists a meromorphic function f M(X ) such that f is holomorphic at all the x i and f (x i ) = a i for 1 i n. Proposition Let φ : Y X be a non-constant holomorphic map of compact connected Riemann surfaces which has degree d as a branched cover. The induced field extension M(Y ) φ M(X ) is finite of degree d.

49 Main theorem The functor Y M(Y ) is a contravariant functor from the category of compact connected Riemann surfaces mapping holomorphically onto a connected compact Riemann surface X to the category of field extensions of M(X ).

50 Main theorem The functor Y M(Y ) is a contravariant functor from the category of compact connected Riemann surfaces mapping holomorphically onto a connected compact Riemann surface X to the category of field extensions of M(X ). Theorem The above functor is an anti-equivalence of categories. In this anti-equivalence, finite Galois branched covers of X correspond to finite Galois extensions of M(X ) of the same degree.

51 Outline 1 The problem 2 Compact Riemann Surfaces 3 Covering Spaces 4 Connection to field theory 5 Proof of the Main theorem 6 References

52 Theorem Every finite group occurs as the Galois group of some finite Galois extension of C(t). Proof. 1 M(P 1 (C)) = C(t).

53 Theorem Every finite group occurs as the Galois group of some finite Galois extension of C(t). Proof. 1 M(P 1 (C)) = C(t). 2 π 1 (P 1 (C) \ {x 1,..., x n }) for some n points x 1,..., x n P 1 (C) is the free group on n 1 letters.

54 Theorem Every finite group occurs as the Galois group of some finite Galois extension of C(t). Proof. 1 M(P 1 (C)) = C(t). 2 π 1 (P 1 (C) \ {x 1,..., x n }) for some n points x 1,..., x n P 1 (C) is the free group on n 1 letters. 3 Let G be a finite group that is the quotient of the free group on n 1 letters. There is a Galois branched cover Y of P 1 (C), branched over the n points x 1,..., x n such that Aut(Y X ) = G.

55 Theorem Every finite group occurs as the Galois group of some finite Galois extension of C(t). Proof. 1 M(P 1 (C)) = C(t). 2 π 1 (P 1 (C) \ {x 1,..., x n }) for some n points x 1,..., x n P 1 (C) is the free group on n 1 letters. 3 Let G be a finite group that is the quotient of the free group on n 1 letters. There is a Galois branched cover Y of P 1 (C), branched over the n points x 1,..., x n such that Aut(Y X ) = G. 4 Let L = M(Y ). This is the field extension we are seeking. By the previous theorem, L is a Galois extension of C(t) and G = Aut(Y X ) = Gal(M(Y ) M(P 1 (C))) = Gal(L C(t))

56 Theorem Every finite group occurs as the Galois group of some finite Galois extension of C(t). Proof. 1 M(P 1 (C)) = C(t). 2 π 1 (P 1 (C) \ {x 1,..., x n }) for some n points x 1,..., x n P 1 (C) is the free group on n 1 letters. 3 Let G be a finite group that is the quotient of the free group on n 1 letters. There is a Galois branched cover Y of P 1 (C), branched over the n points x 1,..., x n such that Aut(Y X ) = G. 4 Let L = M(Y ). This is the field extension we are seeking. By the previous theorem, L is a Galois extension of C(t) and G = Aut(Y X ) = Gal(M(Y ) M(P 1 (C))) = Gal(L C(t)) 5 Every finite group is the quotient of such a free group, and we are done!

57 Outline 1 The problem 2 Compact Riemann Surfaces 3 Covering Spaces 4 Connection to field theory 5 Proof of the Main theorem 6 References

58 References Fundamental Groups and Galois Groups, Támas Szamuely, Cambridge Studies in Advanced Mathematics 117 Lectures on Riemann Surfaces, Otto Forster, GTM 81

59 Thank you! Finally...