Download 3-Selmer groups for curves y^2 = x^3 + a by Bandini A. PDF

By Bandini A.

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We shall show by induction on the degree that every homogeneous polynomial P ∈ J is a (homogeneous) polynomial in I1 , I2 , . . , Ir . This is certainly true for polynomials of degree 0 and so we shall suppose that the degree of P is at least 1. In this case we have P ∈ J + ⊆ F and therefore we may write P = P1 I1 + P2 I2 + · · · + Pr Ir for some P1 , P2 , . . , Pr ∈ S. In addition, for all i, we may choose Pi to be homogeneous of degree deg P − deg Ii . It follows that P = Av(P ) = Av(P1 )I1 + Av(P2 )I2 + · · · + Av(Pr )Ir .

As an application of this result we have the following theorem. 15. Let G be a finite group acting on V = Cn and let S = S(V ∗ ). Let {F1 , . . , Fn } be a set of algebraically independent elements of S G such that S G is integral over C[F1 , . . , Fn ]. Then the map ϕ : V → Cn given by v → (F1 (v), . . , Fn (v)) is surjective. Proof. Since F1 , . . , Fn are algebraically independent, the homomorphism ϕ∗ : C[X1 , . . , Xn ] → S defined above is injective, so we may identify C[X1 , . . , Xn ] with C[F1 , .

5. Suppose that v, w ∈ V . Then there exists g ∈ G such that g(v) = w if and only if P (v) = P (w) for all P ∈ J. 42 3. Polynomial invariants Proof. If g(v) = w, then certainly P (w) = P (g(v)) = P (v). 3 there exists Q ∈ S such that Q(gv) = 1 for each element g ∈ G while Q(w) = 0. Let P = g ∈G gQ. Then P ∈ J = S G , and P (v) = g ∈G Q(g −1 v) = 1, while P (w) = 0. Thus distinct G-orbits are separated by J. 3. Invariants of a finite group Hilbert’s Fourteenth Problem (see Mumford [168]) asks whether the algebra of invariants J of an arbitrary group G is finitely generated as an algebra.

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