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ZHANG’S CONJECTURE AND THE EFFECTIVE BOGOMOLOV CONJECTURE OVER FUNCTION FIELDS
, 2009
"... We prove the Effective Bogomolov Conjecture, and so the Bogomolov Conjecture, over a function field of characteristic 0 by proving Zhang’s Conjecture about certain invariants of metrized graphs. In the function field case, these conjectures were previously known to be true only for curves of genus ..."
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We prove the Effective Bogomolov Conjecture, and so the Bogomolov Conjecture, over a function field of characteristic 0 by proving Zhang’s Conjecture about certain invariants of metrized graphs. In the function field case, these conjectures were previously known to be true only for curves of genus at most 4 and a few other special cases. We also either verify or improve the previous results. We relate the invariants involved in Zhang’s Conjecture to the tau constant of metrized graphs. Then we use and extend our previous results on the tau constant. By proving another Conjecture of Zhang, we obtain a new proof of the slope inequality for Faltings heights on moduli space of curves.
METRIC PROPERTIES OF THE TROPICAL ABELJACOBI MAP
"... Abstract. Let Γ be a tropical curve (or metric graph), and fix a base point p ∈ Γ. We define the Jacobian group J(G) of a finite weighted graph G, and show that the Jacobian J(Γ) is canonically isomorphic to the direct limit of J(G) over all weighted graph models G for Γ. This result is useful for r ..."
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Abstract. Let Γ be a tropical curve (or metric graph), and fix a base point p ∈ Γ. We define the Jacobian group J(G) of a finite weighted graph G, and show that the Jacobian J(Γ) is canonically isomorphic to the direct limit of J(G) over all weighted graph models G for Γ. This result is useful for reducing certain questions about the AbelJacobi map Φp: Γ → J(Γ), defined by Mikhalkin and Zharkov, to purely combinatorial questions about weighted graphs. We prove that J(G) is finite if and only if the edges in each 2connected component of G are commensurable over Q. As an application of our direct limit theorem, we derive some local comparison formulas between ρ and Φ ∗ p (ρ) for three different natural “metrics ” ρ on J(Γ). One of these formulas implies that Φp is a tropical isometry when Γ is 2edgeconnected. Another shows that the canonical measure µZh on a metric graph Γ, defined by S. Zhang, measures lengths on Φp(Γ) with respect to the “supnorm ” on J(Γ). 1.
Admissible constants for genus 2 curves
"... Abstract. S.W. Zhang recently introduced a new adelic invariant ϕ for curves of genus at least 2 over number fields and function fields. We calculate this invariant when the genus is equal to 2. 1. ..."
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Abstract. S.W. Zhang recently introduced a new adelic invariant ϕ for curves of genus at least 2 over number fields and function fields. We calculate this invariant when the genus is equal to 2. 1.