On a conjecture of Izhboldin on similarity of quadratic forms

Abstract

In his paper Motivic equivalence of quadratic forms, Izhboldin modifies a conjecture of Lam and asks whether two quadratic forms, each of which isomorphic to the product of an Albert form and a $k$-fold Pfister form, are similar provided they are equivalent modulo $I^{k+3}$. We relate this conjecture to another conjecture on the dimensions of anisotropic forms in $I^{k+3}$. As a consequence, we obtain that Izhboldin's conjecture is true for $k\le 1$.