[Paper Review] Second variation of Zhang's lambda-invariant on the moduli space of curves

Robin de Jong|arXiv (Cornell University)|Feb 8, 2010

Algebraic Geometry and Number Theory21 references4 citations

TL;DR

This paper computes the second variation of Zhang's λ-invariant on the moduli space of curves, proving that $(8g+4)λ$ is equal to the Hain-Reed β-invariant up to a genus-dependent constant. Using Kawazumi's work on second variations and properties of the Ceresa cycle, the authors derive an explicit formula for the λ-invariant on hyperelliptic curves in terms of the Petersson norm of the discriminant modular form, establishing a deep link between arithmetic invariants and automorphic forms.

ABSTRACT

We compute the second variation of the λ-invariant, recently introduced by S. Zhang, on the complex moduli space M_g of curves of genus g>1, using work of N. Kawazumi. As a result we prove that (8g+4)λis equal, up to a constant, to the β-invariant introduced some time ago by R. Hain and D. Reed. We deduce some consequences; for example we calculate the λ-invariant for each hyperelliptic curve, expressing it in terms of the Petersson norm of the discriminant modular form.

Motivation & Objective

• To compute the second variation of Zhang's λ-invariant on the moduli space $\mathcal{M}_g$ of curves of genus $g \geq 2$.
• To relate the λ-invariant to the Hain-Reed β-invariant via second variation techniques.
• To compute the λ-invariant explicitly for hyperelliptic curves, expressing it in terms of the Petersson norm of the discriminant modular form.
• To establish a precise relationship between arithmetic invariants and automorphic forms on the moduli space.

Proposed method

• Utilizes Kawazumi's second variation formula for the φ-invariant, which is related to Zhang's λ-invariant.
• Applies the Hain-Reed construction of the β-invariant via the biextension line bundle and normal function associated to the Ceresa cycle.
• Compares the second variations of the λ-invariant and the β-invariant to establish their proportionality.
• Uses the fact that the Ceresa cycle vanishes on hyperelliptic curves to deduce that the biextension metric is constant on the hyperelliptic locus.
• Applies the Hodge metric and biextension metric on the determinant of the Hodge bundle $\mathcal{L}^{\otimes 8g+4}$ to compare norms.
• Employs Yamaki's non-archimedean formula for the ε-invariant and the local Cornalba-Harris equality to compute the constant in the λ-β relation.

Experimental results

Research questions

• RQ1 How does the second variation of Zhang's λ-invariant relate to known invariants on the moduli space of curves?
• RQ2 Is the λ-invariant on hyperelliptic curves expressible in terms of classical modular forms?
• RQ3 Can the Hain-Reed β-invariant be directly related to the λ-invariant via second variation?
• RQ4 What is the precise constant of proportionality between $(8g+4)\lambda$ and the β-invariant?
• RQ5 Does the constant in the relation $(8g+4)\lambda = \beta + C_g$ vanish for hyperelliptic curves?

Key findings

• The second variation of Zhang's λ-invariant is computed via Kawazumi's formula, linking it to canonical 2-forms on the universal curve.
• The equality $(8g+4)\lambda = \beta$ holds up to a constant depending only on $g$, establishing a direct link between the λ-invariant and the Hain-Reed β-invariant.
• For hyperelliptic curves, the λ-invariant is given by $(8g+4)\lambda = -\log\|\Lambda\|_{\mathrm{Hdg}}$, where $\Lambda$ is a global trivializing section of $\mathcal{L}^{\otimes 8g+4}$.
• The constant in the relation $(8g+4)\lambda = \beta + C_g$ vanishes for hyperelliptic curves, as shown via non-archimedean analysis and the local Cornalba-Harris equality.
• The λ-invariant on hyperelliptic curves is expressed in terms of the Petersson norm of the discriminant modular form $\Delta_g$, via $\|\Lambda\|_{\mathrm{Hdg}}^{n} = (2\pi)^{4g^2r}\|\Delta_g\|^{g}$.
• The formula $(8g+4)\lambda = g\xi_0 + \sum_{j=1}^{[(g-1)/2]} 2(j+1)(g-j)\xi_j + \sum_{i=1}^{[g/2]} 4i(g-i)\delta_i$ computes the λ-invariant in terms of the types and subtypes of singular fibers in a semi-stable model.