## Representation masses of spinor genera.(English)Zbl 1021.11008

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)], 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.

### MSC:

 1.1e+13 Quadratic forms over global rings and fields 1.1e+09 Quadratic forms over local rings and fields 1.1e+26 Sums of squares and representations by other particular quadratic forms

### Keywords:

integral quadratic form; representation mass; spinor genus

Zbl 0533.10016
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