A Lehmer-type height lower bound for abelian surfaces over function fields

Let K be a 1-dimensional function field over an algebraically closed field of characteristic 0, and let A/K be an abelian surface. Under mild assumptions, we prove a Lehmer-type lower bound for points in A(K̄). More precisely, we prove that there are constants C1, C2 > 0 such that the normalized Bernoulli-part of the canonical height is bounded below by ĥBA(P ) ≥ C1 [ K(P ) : K ]−2 for all points P ∈ A(K̄) whose height satisfies 0 < ĥA(P ) ≤ C2.

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