Jordan splittings of almost unimodular integral quadratic forms

Necessary and sufficient conditions for the existence of such splittings in the indefinite case are given in [Hambleton and Riehm, Invent. Math.45 (1978), 19–33] so we restrict ourselves mainly to the definite case in fact also to those forms which are represented by a unimodular form of the same rank. In this context a natural equivalence relation on such forms is related to a problem over a finite field, and this in turn is investigated more thoroughly in the case when the unimodular representing form is Σ_{i = 1}ⁿX_i²: the number of equivalence classes is counted for small values of n, and it is shown that very few forms have Jordan splittings over $\text{Z}$. A calculation of the Grothendieck and Witt groups of almost unimodular forms is also given.