Rabinowitsch times six

Abstract

We give an analogue of the Rabinowitsch criterion with $\mathbb{Z} $ replaced by the polynomial ring $k[t]$ over a field of characteristic different from $2$. In fact, we expose three different proofs of the Rabinowitsch criterion, using Dedekind-Hasse norms, binary quadratic forms and the Minkowski bound on ideal classes, and adapt each to prove our Polynomial Rabinowitsch criterion. Whereas there are precisely seven cases in which the classical Rabinowitsch criterion holds, working over an arbitrary ground field gives us much more latitude, e.g. recent results about genus $1$ curves yield infinitely many instances in which the Rabinowitsch criterion is satisfied over $k = \mathbb{Q} $. Finally, we take a geometric perspective and relate the Rabinowitsch criterion to the Mordell-Weil group of the Jacobian of the associated hyperelliptic curve.