Indecomposable quadratic spaces over the affine plane

Abstract

Let K be a field of characteristic f2. It is, by now, a well-known theorem of Harder [4, p. 1861 that any quadratic space over the polynomial ring K[X] in one variable is extended from K and hence is decomposable. It is also known that quadratic spaces of ranks 62 and isotropic quadratic spaces of all ranks over K[X,,..., A’,] are extended from K for any rn and hence are again decomposable [8, Proposition 1.1; 5; 111. However, if K admits of a quaternion division algebra, then there do exist indecomposable anisotropic quadratic spaces of ranks 3 and 4 over K[X, Y] [3, Sect. 63. Recently, indecomposable positive definite quadratic spaces of rank 4n over rW[X, Y] were constructed, for each n 3 1 [6], using a classification of these spaces in terms of linear algebraic data. These results motivate the following question: Let K be a field of characteristic 22 which admits of an anisotropic quadratic space q of rank 3 3. Does there exist an indecomposable quadratic space over K[X, Y] whose reduction modulo the variables is isometric to q? The aim of this paper is to answer this question in the affirmative. We in fact prove that if ( A1 ,..., A,, ) is an anisotropic quadratic space of rank 3 3 over K, then there exist infinitely many inequivalent indecomposable quadratic spaces over K[X, Y] whose reductions modulo (A’, Y) are isometric to (i r ,..., i,). The idea of the proof is to start with rank 3 indecomposable quadratic spaces and to build inductively higher rank spaces by the process of glueing. (For a construction of a rank 6 indecomposable quadratic space over lR[X, Y] using a glueing technique, see [2, Sect. 31). The indecomposability is achieved thanks to the rigidity properties of isometries of anisotropic spaces.