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The BSD conjecture for an abelian variety $A$ over a function field holds if $III(A)[\ell^\infty]$ is finite for some prime $\ell$ ($\ell = p$ allowed). This is a theorem by Schneider, Bauer and Kato-Trihan. If $A$ is a constant abelian variety, $III(A)$ is finite by Milne's PhD thesis.