Minimal visualizing abelian varieties for a given nontrivial Sha element can have different dimensions, with explicit minimal genus-2 Jacobians constructed for order-2 elements and computational evidence for order-3 elements.
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On the Visibility category of the Shafarevich--Tate group
Minimal visualizing abelian varieties for a given nontrivial Sha element can have different dimensions, with explicit minimal genus-2 Jacobians constructed for order-2 elements and computational evidence for order-3 elements.