Reflection theorem for divisor class groups of relative quadratic function fields

We present the reflection theorem for divisor class groups of relative quadratic function fields. Let K be a global function field with constant field Fq. Let L1 be a quadratic geometric extension of K and let L2 be its twist by the quadratic constant field extension of K. We show that for every odd integer m that divides q+1 the divisor class groups of L1 and L2 have the same m-rank.