On the vector bundles whose endomorphisms yield Azumaya algebras of cyclic type (Q1115494)

L and M being two n-torsion line bundles over a non-singular quasi- projective variety X over an algebraically closed field of any characteristic, an associative Azumaya algebra A(L,M) of rank \(n^ 2\) is constructed, the elements of which correspond to projective space bundles of rank n over X. It is isomorphic to \({\mathcal E}nd(V)\) for some vector bundle \(V(L,M)\) uniquely determined by \(A(L,M)\) up to tensoring with line bundles, whereas the cup-product \(L\cup M\) is the image in the étale cohomology set \(H^ 2(X,\mu_ n)\) of some line bundle Z under the cycle map. Specializing to the case where X is a product of two elliptic curves and \(n=2\), the \(V(L,M)\)'s are classified and constructed from the Z's, examples are given and there is an application to the existence of rational points on X where the norm of a quaternion Azumaya algebra over X vanishes.