EICHLER ORDERS, TREES AND REPRESENTATION FIELDS

Abstract

The spinor class field for a genus of orders of maximal rank in a quaternion algebra 𝔄 over a number field K is an abelian extension Σ/K provided with a distance function associating elements of the corresponding Galois group to pairs of orders in that genus. If ℌ ⊆ 𝔇 are two orders in a quaternion algebra 𝔄 with 𝔇 of maximal rank, the representation field F = F(𝔇 | ℌ) is a subfield of the spinor class field for the genus of 𝔇 such that, the set of spinor genera of orders in that genus representing the order ℌ, coincides with the set of spinor genera of orders whose distance to 𝔇 fixes F pointwise. Previous works have focused on two cases: maximal orders 𝔇 and commutative orders ℌ. In this work, we give a method to compute the representation field F(𝔇|ℌ) when 𝔇 is the intersection of a finite family of maximal orders, e.g., an Eichler order, and ℌ is arbitrary. Examples are provided.