[Paper Review] Degeneration of Riemann theta functions and of the Zhang-Kawazumi invariant with applications to a uniform Bogomolov conjecture

Robert Wilms|arXiv (Cornell University)|Jan 11, 2021

Algebraic Geometry and Number Theory28 references4 citations

TL;DR

This paper establishes the degeneration behavior of the Riemann theta function and the Zhang–Kawazumi invariant in families of principally polarized abelian varieties over the punctured unit disk, showing that the invariant $\varphi(M_t)$ degenerates as $-\varphi(\Gamma)\log|t|$, where $\Gamma$ is the associated metrized reduction graph. This leads to a uniform positive lower bound for the essential minimum of the Néron–Tate height on tautological cycles of Jacobians over number fields.

ABSTRACT

In this paper we study the degeneration behavior of the norm of the Riemann $θ$-function in a family of principally polarized abelian varieties over the punctured complex unit disc in terms of the associated polarized real torus. As an application, we obtain the degeneration behavior of the Zhang--Kawazumi invariant $φ(M_t)$ of a family of Riemann surfaces $M_t$ in terms of Zhang's invariant $φ(Γ)$ of the associated metrized reduction graph $Γ$. This allows us to deduce a uniform lower bound for the essential minimum of the Néron-Tate height on the tautological cycles of any Jacobian variety over a number field.

Motivation & Objective

• To understand the asymptotic behavior of the norm of the Riemann theta function in degenerating families of principally polarized abelian varieties.
• To relate the degeneration of the Zhang–Kawazumi invariant $\varphi(M_t)$ to the tropical invariant $\varphi(\Gamma)$ of the associated metrized reduction graph $\Gamma$.
• To derive a uniform lower bound for the essential minimum of the Néron–Tate height on tautological cycles of Jacobian varieties over number fields.
• To extend tropical and Arakelov-theoretic invariants to arithmetic geometry, particularly in the context of the uniform Bogomolov conjecture.
• To establish a precise asymptotic formula for the invariant $I(A,\Theta)$ in terms of the tropical moment $I(\Sigma_f)$ and logarithmic terms in $|t|$.

Proposed method

• Analyzes the degeneration of the $L^2$-norm of the Riemann theta function $\|\theta\|^2$ on a family $\mathscr{A}_t \to \Delta^*$ of principally polarized abelian varieties.
• Relates the degeneration of $\|\theta\|^2$ to the tropical Riemann theta function $\|\Psi\|$ on the associated polarized real torus $\Sigma_f$.
• Uses integration over a fixed fundamental domain $[0,1)^{2g}$ equipped with Haar measure to avoid domain dependence on $t$.
• Applies the formula $I(A,\Theta) = \log \int_A \|\theta\|^2 \frac{\nu^g}{g!} - \int_A \log\|\theta\|^2 \frac{\nu^g}{g!}$ to derive asymptotic behavior.
• Combines the asymptotic formula for $I(A,\Theta)$ with known relations between $\varphi$, $\delta$, and $I$ to deduce the degeneration of $\varphi(M_t)$.
• Uses the arithmetic Noether formula and bounds on Néron–Tate heights via adelic metrics on line bundles to establish uniform lower bounds.

Experimental results

Research questions

• RQ1 How does the norm of the Riemann theta function degenerate in a family of principally polarized abelian varieties over the punctured unit disk?
• RQ2 What is the asymptotic behavior of the Zhang–Kawazumi invariant $\varphi(M_t)$ as the family degenerates?
• RQ3 Can the degeneration of $\varphi(M_t)$ be expressed in terms of the tropical invariant $\varphi(\Gamma)$ of the associated metrized reduction graph $\Gamma$?
• RQ4 Does the degeneration behavior lead to a uniform lower bound for the essential minimum of the Néron–Tate height on tautological cycles of Jacobians?
• RQ5 How do the invariants $I(A,\Theta)$, $\delta$, and $\varphi$ interact in the degeneration limit, and what is their arithmetic significance?

Key findings

• The invariant $I(\mathscr{A}_t)$ satisfies $I(\mathscr{A}_t) \sim -I(\Sigma_f)\log|t| - \frac{\dim\Sigma_f}{2}\log(-\log|t|)$, describing its degeneration in terms of the tropical moment.
• The Zhang–Kawazumi invariant degenerates as $\varphi(M_t) \sim -\varphi(\Gamma)\log|t|$, where $\Gamma$ is the metrized reduction graph of the degenerating family.
• This asymptotic formula generalizes earlier results by de Jong and provides a tropical explanation for the archimedean behavior of $\varphi$.
• A uniform lower bound for the essential minimum of the Néron–Tate height on tautological cycles is established: $h'_{\mathcal{L}}(Z_{m,\alpha}) \geq \frac{(g-1)^3}{24(47g^4 + 42g^3 + 18g^2 + g)} \max(12h_{\mathrm{Fal}}(X) + c_1(g), c_2(g))$.
• The bound is independent of the choice of cycle parameters $m$ and $\alpha$, and holds uniformly across all number fields.
• The result confirms a uniform version of the Bogomolov conjecture for tautological cycles in Jacobians, extending previous results to higher-dimensional cycles.