Comparing Arithmetic Intersection Formulas for Denominators of Igusa Class Polynomials (2013)

Bruinier and Yang conjectured a formula for intersection numbers on an arithmetic Hilbert modular surface, and as a consequence obtained a conjectural formula for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C upper M left-parenthesis upper K right-parenthesis period normal upper G 1"> <mml:semantics> <mml:mrow> <mml:mi>CM</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>.</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">G</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {CM}(K).\mathrm {G}_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> under strong assumptions on the ramification in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Yang later proved this conjecture under slightly stronger assumptions on the ramification. In recent work, Lauter and Viray proved a different formula for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C upper M left-parenthesis upper K right-parenthesis period normal upper G 1"> <mml:semantics> <mml:mrow> <mml:mi>CM</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>.</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">G</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {CM}(K).\mathrm {G}_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for primitive quartic CM fields with a mild assumption, using a method of proof independent from that of Yang. In this paper we show that these two formulas agree, for a class of primitive quartic CM fields which is slightly larger than the intersection of the fields considered by Yang and Lauter and Viray. Furthermore, the proof that these formulas agree does <italic>not</italic> rely on the results of Yang or Lauter and Viray. As a consequence of our proof, we conclude that the Bruinier-Yang formula holds for a slightly largely class of quartic CM fields <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> than what was proved by Yang, since it agrees with the Lauter-Viray formula, which is proved in those cases. The factorization of these intersection numbers has applications to cryptography: precise formulas for them allow one to compute the denominators of Igusa class polynomials, which has important applications to the construction of genus <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> curves for use in cryptography.