[Paper Review] Singular moduli refined

Benjamin Howard, Tonghai Yang|arXiv (Cornell University)|Feb 28, 2012

Algebraic Geometry and Number Theory18 references3 citations

TL;DR

This paper refines Gross and Zagier's work on singular moduli by introducing an $oscriptstyle{\mathcal{O}}_F$-quadratic form $\deg_{\mathrm{CM}}$ on $\mathrm{Hom}(E_1,E_2)$ for CM elliptic curves $E_1, E_2$ with complex multiplication by orders in imaginary quadratic fields $K_1, K_2$ of relatively prime discriminants. It proves that the arithmetic degree of the moduli stack $\mathcal{X}_\alpha$ parameterizing pairs $(E_1,E_2,j)$ with $\deg_{\mathrm{CM}}(j) = \alpha$ equals the $\alpha$-th Fourier coefficient of the central derivative of a Hecke Eisenstein series of weight 1 over the real quadratic field $F = \mathbb{Q}(\sqrt{d_1d_2})$, establishing an arithmetic Siegel-Weil formula.

ABSTRACT

We prove a refinement of the results of Gross and Zagier on prime factorizations of singular moduli.

Motivation & Objective

• To extend Gross and Zagier's singular moduli results beyond the diagonal case by studying the central derivative of a Hilbert modular Eisenstein series of weight 1.
• To define a canonical $\mathcal{O}_F$-quadratic form $\deg_{\mathrm{CM}}$ on $\mathrm{Hom}(E_1,E_2)$ that is totally positive definite and satisfies $\mathrm{Tr}_{F/\mathbb{Q}}\deg_{\mathrm{CM}} = \deg$.
• To establish an arithmetic Siegel-Weil formula by relating the arithmetic degree of the moduli stack $\mathcal{X}_\alpha$ to the $\alpha$-th Fourier coefficient of the central derivative of the Eisenstein series.
• To resolve the open question of the arithmetic meaning of the $\alpha$-th Fourier coefficient of $E^{*,\prime}(\tau_1,\tau_2,0)$ for $\alpha \in F^\times$.

Proposed method

• Construct a Hecke Eisenstein series $E^*(\tau_1,\tau_2,s)$ of parallel weight 1 for $\mathrm{SL}_2(\mathcal{O}_F)$, which vanishes at $s=0$ due to its functional equation.
• Define a canonical $\mathcal{O}_F$-module structure and a totally positive definite $\mathcal{O}_F$-quadratic form $\deg_{\mathrm{CM}}$ on $\mathrm{Hom}(E_1,E_2)$ using the CM actions on $E_1$ and $E_2$.
• Introduce the moduli stack $\mathcal{X}_\alpha$ parameterizing triples $(E_1,E_2,j)$ with $\deg_{\mathrm{CM}}(j) = \alpha$, and show it has dimension 0.
• Prove that the arithmetic degree of $\mathcal{X}_\alpha$ equals the $\alpha$-th Fourier coefficient of $E^{*,\prime}(\tau_1,\tau_2,0)$ via local calculations at primes $\mathfrak{p}$ dividing $\mathrm{Diff}(\alpha)$, using Whittaker functions and orbital integrals.
• Use the functional equation and analytic continuation of the Eisenstein series to relate the central derivative to arithmetic invariants of $\mathcal{X}_\alpha$.
• Establish the formula $4\deg(\mathcal{X}_\alpha) \cdot q^\alpha = E_{\alpha}^{*,\prime}(\tau,0)$ when $\mathrm{Diff}(\alpha) = \{\mathfrak{p}\}$, and show both sides vanish when $|\mathrm{Diff}(\alpha)| > 1$.

Experimental results

Research questions

• RQ1 What is the arithmetic interpretation of the $\alpha$-th Fourier coefficient of the central derivative $E^{*,\prime}(\tau_1,\tau_2,0)$ of the Hecke Eisenstein series for $\alpha \in F^\times$?
• RQ2 Can the moduli stack $\mathcal{X}_\alpha$ parameterizing $j \in \mathrm{Hom}(E_1,E_2)$ with $\deg_{\mathrm{CM}}(j) = \alpha$ be used to define an arithmetic invariant matching the Fourier coefficient of $E^{*,\prime}$?
• RQ3 How does the $\mathcal{O}_F$-quadratic form $\deg_{\mathrm{CM}}$ on $\mathrm{Hom}(E_1,E_2)$ refine the standard degree form and relate to the trace to $\mathbb{Q}$?
• RQ4 What is the role of the different $\mathfrak{D}$ of $F/\mathbb{Q}$ in the local structure of $\mathcal{X}_\alpha$ and the arithmetic degree computation?
• RQ5 Under what conditions does $\mathcal{X}_\alpha$ have positive arithmetic degree, and how does this relate to the splitting behavior of $\alpha$ in $F$?

Key findings

• The moduli stack $\mathcal{X}_\alpha$ parameterizing triples $(E_1,E_2,j)$ with $\deg_{\mathrm{CM}}(j) = \alpha$ has dimension 0 for all $\alpha \in F^\times$.
• The arithmetic degree of $\mathcal{X}_\alpha$ is equal to the $\alpha$-th Fourier coefficient of the central derivative $E^{*,\prime}(\tau_1,\tau_2,0)$ of the Hecke Eisenstein series over $F$.
• When $\mathrm{Diff}(\alpha) = \{\mathfrak{p}\}$, the arithmetic degree satisfies $4\deg(\mathcal{X}_\alpha) \cdot q^\alpha = E_{\alpha}^{*,\prime}(\tau,0)$, establishing the arithmetic Siegel-Weil formula.
• If $|\mathrm{Diff}(\alpha)| > 1$, then $\mathcal{X}_\alpha$ is empty and both sides of the arithmetic Siegel-Weil formula vanish.
• The local contribution to the arithmetic degree at a prime $\mathfrak{p}$ is governed by $\nu_{\mathfrak{p}}(\alpha) = \frac{1}{2}\mathrm{ord}_{\mathfrak{p}}(\alpha\mathfrak{p}\mathfrak{D})$, which appears in the derivative of the Whittaker function.
• The canonical $\mathcal{O}_F$-quadratic form $\deg_{\mathrm{CM}}$ is totally positive definite and satisfies $\mathrm{Tr}_{F/\mathbb{Q}}\deg_{\mathrm{CM}} = \deg$, refining the standard degree on $\mathrm{Hom}(E_1,E_2)$.