On embedding numbers into quaternion orders
Let \(R\) be a complete discrete valuation ring of characteristic different from 2 with quotient field \(K\). If \(S\) is an \(R\)-order in a quadratic extension \(L\) of \(K\) and \(\Lambda\) is an \(R\)-order in a quaternion algebra \(A\) over \(K\), an embedding \(S\to \Lambda\) is called optimal if its cokernel is a projective \(R\)-module. The primary aim of this paper is to determine the number of such optimal embeddings modulo the action of the unit group \(\Lambda^*\) by conjugation, when \(\Lambda\) is an Eichler order (and \(S\) is arbitrary). The main result is an explicit formula which yields this number in terms of the number of elements in the residue field of \(R\) (which is assumed to be finite), the discriminant of the Eichler order \(\Lambda\) and the conductor of \(S\) in the maximal order of \(L\), generalizing earlier computations due to Eichler and Hijikata. From this result, the author derives a stability property for embedding numbers, showing that for a given Eichler order \(\Lambda\) and a given quadratic extension \(L/K\), the number of optimal embeddings of an order \(S\) into \(\Lambda\) modulo the action of \(\Lambda^*\) does not depend on \(S\), provided its conductor is relatively small and the Gorenstein closure of \(\Lambda\) is a Bass order. In the final section, applications of embedding numbers to representation of integers by ternary quadratic forms are discussed.
representation of integers by ternary quadratic forms
scientific article; zbMATH DE number 4212173