A relation between embedding degrees and class numbers of binary quadratic forms

Abstract

<p>In this paper, we describe a relation between the embedding degree of an elliptic curve over a prime field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper F Subscript p"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbb {F}_p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the inertial degree of the primes above <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in a certain ring class field. From this relation, we conclude that the embedding degree divides the class number of a group of binary quadratic forms of a fixed discriminant.</p>