[Paper Review] Jumps and monodromy of abelian varieties

Lars Halvard Halle, Johannes Nicaise|arXiv (Cornell University)|Sep 20, 2010

Algebraic Geometry and Number Theory22 references4 citations

TL;DR

This paper proves a strong form of the motivic monodromy conjecture for abelian varieties by establishing that the order of the unique pole of the motivic zeta function equals the maximal rank of a Jordan block of the corresponding monodromy eigenvalue on the degree $g$ cohomology. It further provides a Hodge-theoretic interpretation of Edixhoven's filtration jumps via limit mixed Hodge structures, linking the potential toric rank to monodromy eigenvalues and Jordan block sizes.

ABSTRACT

We prove a strong form of the motivic monodromy conjecture for abelian varieties, by showing that the order of the unique pole of the motivic zeta function is equal to the size of the maximal Jordan block of the corresponding monodromy eigenvalue. Moreover, we give a Hodge-theoretic interpretation of the fundamental invariants appearing in the proof.

Motivation & Objective

• To establish a strong form of the motivic monodromy conjecture for abelian varieties, linking the pole order of the motivic zeta function to the Jordan block structure of monodromy.
• To provide a Hodge-theoretic interpretation of the jumps in Edixhoven’s filtration on the Néron model’s special fiber.
• To relate the potential toric rank $t_{\mathrm{pot}}(A)$ to the monodromy action on $\ell$-adic cohomology and limit mixed Hodge structures.
• To distinguish abelian and dual abelian jumps via Hodge types in the limit mixed Hodge structure when $K = \mathbb{C}((t))$.
• To extend the framework for potential generalization to Calabi-Yau varieties over $\mathbb{C}((t))$.

Proposed method

• Use Edixhoven’s filtration on the special fiber of the Néron model of $A$ to decompose jumps into toric, abelian, and dual abelian types.
• Analyze the monodromy action on $\ell$-adic cohomology via the tame monodromy group $G(K^t/K)$, identifying the action of a topological generator $\sigma$.
• Apply the theory of Néron models of variations of Hodge structures to relate the weight filtration and Hodge structures on $H^g(X_\infty, \mathbb{Q})$.
• Use the Chevalley decomposition of the special fiber of the Néron model to identify the abelian and torus parts of the Néron model via analytification of the Néron model's special fiber.
• Establish canonical $\mu$-equivariant isomorphisms between cohomology groups and Lie algebras of abelian and toric parts of the Néron model.
• Leverage the isomorphism $\mathrm{Gr}_{-1}^W H \cong H_1(B(\mathbb{C}), \mathbb{Z})$ and $\mathrm{Gr}_{-2}^W H \cong H_1(T(\mathbb{C}), \mathbb{Z})$ to relate Hodge structures to monodromy block sizes.

Experimental results

Research questions

• RQ1 Is the order of the unique pole of the motivic zeta function $Z_A(\mathbb{L}^{-s})$ equal to the maximal rank of a Jordan block of the monodromy eigenvalue on $H^g(A, \mathbb{Q}_\ell)$?
• RQ2 How do the jumps in Edixhoven’s filtration on the Néron model’s special fiber relate to the Hodge-theoretic structure of the limit mixed Hodge structure?
• RQ3 Can the potential toric rank $t_{\mathrm{pot}}(A)$ be interpreted as the largest $\alpha$ such that $\exp(2\pi c(A)i)$ is an eigenvalue on $\mathrm{Gr}_{g+\alpha}^W H^g(X_\infty, \mathbb{Q})$?
• RQ4 How do the abelian and dual abelian jumps in Edixhoven’s filtration differ in terms of Hodge type in the limit mixed Hodge structure?
• RQ5 What is the role of the monodromy action on the Tate module in determining the Jordan block structure of the monodromy transformation?

Key findings

• The order of the unique pole of the motivic zeta function $Z_A(\mathbb{L}^{-s})$ at $s = c(A)$ is equal to $1 + t_{\mathrm{pot}}(A)$, and this value equals the maximal rank of a Jordan block of the monodromy eigenvalue on $H^g(A, \mathbb{Q}_\ell)$.
• The potential toric rank $t_{\mathrm{pot}}(A)$ is equal to the largest integer $\alpha$ such that $\exp(2\pi c(A)i)$ is an eigenvalue of the semi-simple part $M_s$ on $\mathrm{Gr}_{g+\alpha}^W H^g(X_\infty, \mathbb{Q})$.
• The Jordan form of $M_s$ on $\mathrm{Gr}_{-1}^W H_1(X_\infty, \mathbb{Q})$ is $\mathrm{Jord}(m_A^{\mathrm{ab}}, 0)$ on the $(1,0)$-part and $\mathrm{Jord}(\breve{m}_A^{\mathrm{ab}}, 0)$ on the $(0,1)$-part, corresponding to the abelian and dual abelian parts.
• The Jordan form on $\mathrm{Gr}_{-2}^W H_1(X_\infty, \mathbb{Q})$ is $\mathrm{Jord}(m_A^{\mathrm{tor}}, 0)$, corresponding to the toric part.
• The canonical $\mu$-equivariant isomorphisms $\mathrm{Gr}_{-1}^W H \cong H_1(B(\mathbb{C}), \mathbb{Z})$ and $\mathrm{Gr}_{-2}^W H \cong H_1(T(\mathbb{C}), \mathbb{Z})$ are compatible with the monodromy action, confirming the Hodge-theoretic interpretation.
• The monodromy action on the cohomology is fully described by the decomposition into weight-graded pieces, with the unipotent part $N = \log M_u$ inducing the weight filtration, and the semi-simple part $M_s$ acting via the Hodge structure.