| Summary: | <p>We give an explicit rational parameterization of the surface <strong><em>H</em><sub>3</sub></strong> over ℚ whose points parameterize genus 2 curves <em>C</em> with full √3-level structure on their Jacobian <em>J</em>. We use this model to construct abelian surfaces <em>A</em> with the property that III (<em>A<sub>d</sub></em>) [3] ≠ 0 for a positive proportion of quadratic twists <em>A<sub>d</sub></em>. In fact, for 100% of <em>x</em>∈<strong><em>H</em><sub>3</sub></strong>(ℚ), this holds for the surface <em>A</em> = Jac(<em>C<sub>x</sub></em>)/⟨<em>P</em>⟩, where <em>P</em> is the marked point of order 3. Our methods also give an explicit bound on the average rank of <em>J<sub>d</sub></em>(ℚ), as well as statistical results on the size of #<em>C<sub>d</sub></em>(ℚ), as <em>d</em> varies through squarefree integers.</p>
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