Applications of a Siegel-like formula of Eichler orders

Abstract

We obtain a Siegel-like formula on all Eichler orders with a same squarefree level in a definite quaternion algebra over the field of rationals. Applying it, we give three unified formulae for the number of elements with a same reduced norm and reduced trace in one of the Eichler orders in three cases respectively, when the type number of the Eichler orders equals one. If these Eichler orders belong to more than one type, using the Jacobi theta series and the Jacobi Eisenstein series in the Siegel-like formula, we construct Jacobi cusp forms which correspond to certain cusp forms of weight 3/2 in Kohnen's plus space by Gross' construction. The latter can help to provide certain newforms of weight 3/2 in Kohnen's plus space whose Shimura lift are the trace forms of certain subspaces of the space of newforms of weight 2. These trace forms are sometimes the newforms of weight 2 corresponding to elliptic curves over the rational numbers with a squarefree conductor, whose modified L-functions have even functional equations.