Class number relations in almost abelian number fields

Abstract This paper explores the leading terms of equivariant Artin L -functions at $$s=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> for finite abelian extensions F / k , where k is an imaginary quadratic number field. We introduce a module constructed from these leading terms and the Dirichlet regulator, and compute its generalized index relative to the torsion-free part of the unit group of F . The main result establishes that this index equals the ratio of class numbers $$h_F/h_k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>h</mml:mi> <mml:mi>F</mml:mi> </mml:msub> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>h</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> </mml:math> multiplied by an explicit annihilator index involving roots of unity. For primes p not dividing the Galois group order, we provide a p -adic refinement using idempotents from higher Stark theory. As consequences, we derive obstructions to the existence of integral higher Stark elements mapping to the equivariant L -value and prove class number divisibility under non-decomposition conditions for ramified places. These findings contribute to the verification of equivariant Stark-type conjectures and offer new insights into class number relations in abelian extensions.