Darstellung durch definite ternäre quadratische Formen

AbstractWe study the representation behaviour of a Z-lattice L on a positive definite ternary quadratic space V over Q. As a new tool for this we use the Bruhat-Tits building of the spingroup of the completion of V at a suitable prime p. In Section 2 we show how this can be described in an elementary way as a graph whose vertices are the Zp-maximal lattices on Vp, and in Section 4 we let this graph induce a graph, whose vertices are lattices on V, which differ from L only at the prime p. In Section 3 we investigate which lattices from the graph defined in Section 2 have a given vector in common. The results are used in Sections 5 and 6 to obtain information on the representation behaviour of some special lattices. In Section 5 we get a list of lattices, which represent all numbers they represent locally everywhere; this list contains that given by Watson in [16]. In Section 6 we sharpen a result of Jones and Pall from [6]