Igusa, J. On a certain class of prehomogeneous vector spaces. J. of Pure and Applied Algebra, 47:265-282, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....G SL(7) t 3 t 1 and so t 1. This proves the proposition. By the above proposition, G SL(7) Aut ( Therefore, Hi( k , w) r Hl(k, Aut ( x Hi(k,T) r Hl(k, Aut ( G V, ss is in bijective correspondence with k forms of . This was proved by Hence, k k Sato Kimura [23] and Igusa [10]. We make this correspondence more explicit. For that purpose, we shall prove the following theorem. Theorem 20.34. The map is well defined and is bijective. Proof. If x, y V s, t, g) G, and y = t, g)x, then mx,y, t,g) is a k isomorphism. Therefore, the above map is well defined. Suppose x, ....

Igusa, J. On a certain class of prehomogeneous vector spaces. J. of Pure and Applied Algebra, 47:265-282, 1987.

....rational orbit decompositions for certain regular prehomogeneous vector spaces in [8] 4] in all the characteristics, and we had to worry about the separability. Because of the above theorem, all we have to do is to make sure that Gw is smooth and reductive, and by exactly the same argument as in [3], we have the above interpretation of G k nV ss k in terms of Galois cohomology sets, and we no longer have to worry about the separability. This was the motivation of this paper. Of course, we still have to show that Gw is reductive, but that is necessary anyway to determine the orbit space G k ....

Igusa, J. On a certain class of prehomogeneous vector spaces. J. of Pure and Applied Algebra, 47:265--282, 1987.

....= 1; g 0 Gamma1 g 0oe ) 2 G w k : So this defines a cohomology class in H 1 (k; Gw ) This implies that x = t; g)w 2 V ss k . Since g 0 = m w;x; t;g) g 0 induces an isomorphism from Im( e O ) k to Im(O x ) k . Therefore, A = O x . Remark (2. 25) It is proved in [7] [4] that G k n V ss k = H 1 (k; Aut ( e O ) So the credit for the existence of a bijective correspondence between G k nV ss k and O should go to Sato Kimura [7] and Igusa [4] However, we constructed O x 2 O for x 2 V ss k , and the fact that this particular correspondence is ....

....an isomorphism from Im( e O ) k to Im(O x ) k . Therefore, A = O x . Remark (2. 25) It is proved in [7] 4] that G k n V ss k = H 1 (k; Aut ( e O ) So the credit for the existence of a bijective correspondence between G k nV ss k and O should go to Sato Kimura [7] and Igusa [4]. However, we constructed O x 2 O for x 2 V ss k , and the fact that this particular correspondence is bijective still required a proof. The operator D 3 was considered in [7] The fact that the stabilizer of w is a group of type G 2 at least goes back to [7] For the rest of this section, we ....

Igusa, J. On a certain class of prehomogeneous vector spaces. J. of Pure and Applied Algebra, 47:265--282, 1987.

....Condition (1.5) is satisfied. Then for any x 2 V ss k , we can choose a finite Galois extension k 0 =k and g 2 G k 0 such that x = gw. Then c x = fg Gamma1 g j g determines an element of Ker(H 1 (k; Gw ) H 1 (k; G) which is the set of elements which map to 1 2 H 1 (k; G) In [2] Igusa assumed that the characteristic of the field is zero. However, if Condition (1.5) is satisfied, we can still make cohomology classes from rational orbits in V ss k . Therefore, without changing Igusa s argument, we have the following Theorem. Theorem (1.6) Igusa) Suppose a ....

Igusa, J. On a certain class of prehomogeneous vector spaces. J. of Pure and Applied Algebra, 47:265--282, 1987.

....considered by many people. For example Gauss [1] considered the space of binary quadratic forms to investigate ideal classes of quadratic fields. Igusa [3] investigated spinors of dimension up to twelve, and these cases turned out to be special cases of prehomogeneous vector spaces. Later Igusa [4] formulated the problem in terms of Galois cohomology, and this formulation was used to parametrize field extensions of degree up to five in [11] Let W = k 7 be the standard representation of GL(7) i.e. the space of seven dimensional column vectors) and e 1 ; Delta Delta Delta ; e 7 the ....

....Galois extension k 0 =k and g 2 G k 0 such that x = gw. Then c x = fg Gamma1 g j g determines an element of Ker(H 1 (k; Gw ) H 1 (k; G) which is the set of elements which map to 1 2 H 1 (k; G) The following theorem (which does not require Assumption (1. 4) is due to Igusa [4]. Theorem (1.6) Igusa) The correspondence G k n V ss x 3 x c x 2 Ker(H 1 (k; Gw ) H 1 (k; G) is bijective. By the above theorem, we can consider G k n V ss k as a subset of H 1 (k; Gw ) We denote the restriction of ff V ; fl V to G k n V ss k also by ff V ; fl V . Let x 2 V ....

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Igusa, J. On a certain class of prehomogeneous vector spaces. J. of Pure and Applied Algebra, 47:265--282, 1987.

....vector space, and w 2 V ss k . Then for any x 2 V ss k , there exists g x 2 G k such that x = g x w. Then c x = fg Gamma1 x g oe x g oe2Gal( k=k) determines a cohomology class in H 1 (k; Gw ) and does not depend on the choice of g x . The following theorem is due to Igusa [8]. Theorem (4.1) Igusa) The correspondence G k n V ss x 3 x c x 2 Ker(H 1 (k; Gw ) H 1 (k; G) is bijective. Note that Ker(H 1 (k; Gw ) H 1 (k; G) is the set of elements c 2 H 1 (k; Gw ) which map to the trivial class in H 1 (k; G) In our case, H 1 (k; G) is trivial. ....

Igusa, J. On a certain class of prehomogeneous vector spaces. J. of Pure and Applied Algebra, 47:265--282, 1987.

....is due to him. x1 Invariant theory of the space 3 k 8 Let G = GL(1) Theta GL(8) V = 3 k 8 , and e T = Ker(G GL(V ) It is easy to see that e T = f(t Gamma3 ; tI 8 ) j t 2 GL(1)g = GL(1) This is the prehomogeneous vector space of type (7) in [31] also discussed in Igusa [17]. Even though this is known to be a prehomogeneous vector space, we have to study the invariant theory of this space for our purposes. Let fe 1 ; Delta Delta Delta ; e 8 g be a basis of k 8 . We use the notation e i 1 Delta Delta Deltai k = e i 1 Delta Delta Delta e i k . It is known ....

....cohomology set. We choose the definition so that trivial classes are those of the form fg Gamma1 g oe g oe2Gal( k=k) g 2 G k ) and the cocycle condition is h oe = h h oe for a continuous map fh oe g oe2Gal( k=k) from Gal( k=k) to G k . Note that by a similar argument as in [17], Gw = GL(3) Z(GL(3) Theta e T . Since H 1 (k; G) H 1 (k; e T ) f1g, the following theorem follows. Theorem (1.5) Sato Kimura, Igusa) G k n V ss k = H 1 (k; Aut (SL(3) H 1 (k; Aut (GL(3) By the above theorem, the orbit space G k n V ss k corresponds ....

Igusa, J. On a certain class of prehomogeneous vector spaces. J. of Pure and Applied Algebra, 47:265--282, 1987.

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