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On a Conjecture of Yui and Zagier II

Yingkun Li, Tonghai Yang, Dongxi Ye

TL;DR

We address Yui–Zagier’s conjectures on the norm of the difference of Weber class invariants for equal discriminants by applying Schofer’s small CM value formula. The authors develop a general main CM-formula that expresses the relevant norms via explicit arithmetic densities and local data, leading to a discriminant case recovery of YZ and a precise resultant factorization for Weber polynomials. The work generalizes LY’s big-CM approach to the small-CM setting, clarifies level-splitting via a Weber-character, and provides detailed local calculations (Whittaker, splitting, and densities) that assemble into the global factorization formulas. This establishes new, explicit links between CM theory, modular functions on Shimura varieties, and arithmetic factorization phenomena with concrete numerical examples illustrating the results.

Abstract

Yui and Zagier made some fascinating conjectures on the factorization on the norm of the difference of Weber class invariants $ f(\mathfrak a_1) - f(\mathfrak a_2)$ based on their calculation in \cite{YZ}. Here $\mathfrak a_i$ belong two diferent ideal classes of discrimants $D_i$ in imagainary quadratic fields $\mathbb{Q}(\sqrt{D_i})$. In \cite{LY}, we proved these conjectures and their generalizations when $(D_1, D_2) =1$ using the so-called big CM value formula of Borcherds lifting. In this sequel, we prove the conjectures when $\mathbb{Q}(\sqrt{D_1}) =\mathbb{Q}(\sqrt{D_2})$ using the so-called small CM value formula. In addition, we give a precise factorization formula for the resultant of two different Weber class invariant polynomials for distinct orders.

On a Conjecture of Yui and Zagier II

TL;DR

We address Yui–Zagier’s conjectures on the norm of the difference of Weber class invariants for equal discriminants by applying Schofer’s small CM value formula. The authors develop a general main CM-formula that expresses the relevant norms via explicit arithmetic densities and local data, leading to a discriminant case recovery of YZ and a precise resultant factorization for Weber polynomials. The work generalizes LY’s big-CM approach to the small-CM setting, clarifies level-splitting via a Weber-character, and provides detailed local calculations (Whittaker, splitting, and densities) that assemble into the global factorization formulas. This establishes new, explicit links between CM theory, modular functions on Shimura varieties, and arithmetic factorization phenomena with concrete numerical examples illustrating the results.

Abstract

Yui and Zagier made some fascinating conjectures on the factorization on the norm of the difference of Weber class invariants based on their calculation in \cite{YZ}. Here belong two diferent ideal classes of discrimants in imagainary quadratic fields . In \cite{LY}, we proved these conjectures and their generalizations when using the so-called big CM value formula of Borcherds lifting. In this sequel, we prove the conjectures when using the so-called small CM value formula. In addition, we give a precise factorization formula for the resultant of two different Weber class invariant polynomials for distinct orders.

Paper Structure

This paper contains 22 sections, 40 theorems, 208 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weil Representation.
  4. Modular curves and product of modular curves as Shimura varieties
  5. CM points on $\mathbb{H}$ as CM points associated to $V^0$
  6. Small CM points on $\mathbb{H} \times \mathbb{H}$
  7. Lattices
  8. Level
  9. Borcherds products and small CM value formula
  10. A brief review of a result of LY
  11. Incoherent Eisenstein Series
  12. Small CM value formula
  13. Statement and Proof of Main Result
  14. The main formula
  15. The Yui-Zagier conjecture on discriminant
  16. ...and 7 more sections

Key Result

Proposition 1.1

Let $D<0$ be a discriminant satisfying Denote $\varepsilon_D := (-1)^{(D-1)/8}$. For each ideal $\mathfrak a=[a, \frac{b+\sqrt D}{2}]$ of the order $\mathcal{O}_D := \mathbb{Z}[\tfrac{D+ \sqrt{D}}{2} ]$ with $a>0$, let $z_{\mathfrak a} =\frac{-b+\sqrt{D}}{2a}$ be the associated CM point and Then $f(\mathfrak a)$ is an algebraic integer depending only on the class of $\mathfrak a$ in the class gr

Theorems & Definitions (81)

  • Proposition 1.1: YZ Proposition
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • ...and 71 more