The zeta function of a class of rational varieties

Abstract

Let k be a field and V be an algebraic fc-scheme of dimension w-1.V is called a Severi-Brauer /;-scheme if there exists a separable algebraic extension L/k such that V x k L and PV is said to be split by L/k.V is called a trivial Severi-Brauer ^-scheme if V and -P w _i(fc) are isomorphic as fcschemes.Let K/k be a finite Galois extension and let G=Gal(X/A:).The isomorphism classes (as /^-schemes) of Severi-Brauer /c-schemes of dimension n-\ which are split by K/k are in canonical one-one correspondence with the elements of the cohomology set H^G.PGLin.K))A is injective and ImA is described as follows: let y e H 2 (G,K*) and let D(y) denote the central division algebra over k defined by y, let [D(y):k]=d 2 , then y e lm A if and only if d\n [6].Assume now that y e lm A and let j8=A~1(y), then the Severi-Brauer A:-scheme F(/9), defined by /?, is isomorphic as a ^-scheme to the Grassmann variety of left ideals of rank n in the matrix algebra A=M n / d (D(y)).Actually, A does not depend on K but only on n and the image of y in Br (A:), the Brauer group of k.It was Chatelet [5] who first defined Severi-Brauer varieties and established their basic properties over fields of characteristic zero.In particular, he determined the function field, K(V((3)), of F(/?).Amitsur [3] interpreted Chatelet's work from another point of view, obtained new results, and extended Chatelet's theorems to fields of arbitrary characteristic.Today, his determination of K(V(f3)) is more accessible than Chatelet's.The defining relations for K(V(f})) are simplest when A is a cyclic algebra [1], [2], i.e. there exists a cyclic splitting field k'/k for A with [k':k]=n.In this case, let zek' such that k'=k(z).

Locations

• Project Euclid (Cornell University) - View - PDF
• Bulletin of the American Mathematical Society - View - PDF