Representation masses of spinor genera (Q1847876)

Let \(f\) be an integral quadratic form of rank \(n\). The Minkowski-Siegel mass formula provides a means of determining the representation measure (or mass) of an integer \(c\) by the genus of \(f\) by expressing this quantity as a product of local factors. This method has subsequently been extended, via a mass formula involving quadratic characters, to study the corresponding representation measures of \(c\) by the spinor genera within the genus of \(f\). When \(n \geq 4\), these measures are equal for all spinor genera in the genus of \(f\). However, when \(n=3\) it can happen that two different values of this measure occur within a genus, each occurring for exactly half of the spinor genera in the genus. In this case, the integer \(c\) is referred to as a splitting integer for the genus. In [J. Reine Angew. Math. 352, 114-132 (1984; Zbl 0533.10016)], \textit{R. Schulze-Pillot} studied the difference between the values of the two measures in such a case, expressing this difference as a product of local factors. The present paper extends much of this previous theory to the general context of representations of a quadratic lattice of arbitrary rank by the spinor genera within a genus of lattices of greater or equal rank, and goes on to investigate the case in which the represented lattice has codimension less than 2. More specifically, let \(F\) be an algebraic number field with ring of integers \(o_F\), let \(L\) be an \(o_F\)-lattice of rank \(n\) on a nondegenerate quadratic space \(V\) over \(F\), and let \(K\) be a sublattice of \(L\) of rank \(k\). The author considers the values taken on by the representation measure \(r(K, \text{spn}^+(M))\), as \(\text{spn}^+(M)\) ranges over the proper spinor genera in the genus gen(\(L\)). The paper is divided into two parts. The first part deals with the codimension 2 case (i.e., \(n-k=2\)). Here \(r(K, \text{spn}^+(M))\) takes on at most two distinct values as \(M\) ranges over the lattices in gen(\(L\)); \(K\) is said to be a splitting lattice for gen(\(L\)) when two distinct values are realized. The formula given by Schulze-Pillot is generalized here to produce an expression as a product of local factors for the difference between the representation measures of \(K\) by two proper spinor genera in gen(\(L\)). Criteria are then given for the determination of the local factors appearing in the resulting formula. In the special case when \(n=3\), these criteria are made completely explicit, thus providing a means for determining the splitting integers for a ternary genus. The second part of the paper treats the case when the codimension is less than or equal to one. The quadratic characters on a suitable factor group of the idèle group of \(F\) are used to produce a mass formula for spinor genera in this case. It is no longer true that there are at most two distinct values for \(r(K,\text{spn}^+(M))\) within gen(\(L\)). The author obtains an upper bound for the number of different values that this quantity can take on within a given genus, expressed as a group index involving local spinor norm groups.