Quadratic twists as random variables

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D not-equals 1"> <mml:semantics> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo> ≠ </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">D\neq 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a fixed squarefree integer. For elliptic curves <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E slash double-struck upper Q"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">E/\mathbb {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , writing <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E Subscript upper D"> <mml:semantics> <mml:msub> <mml:mi>E</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>D</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">E_{D}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for the quadratic twist by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we consider the question of how often <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E left-parenthesis double-struck upper Q right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">E(\mathbb {Q})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E Subscript upper D Baseline left-parenthesis double-struck upper Q right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>D</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">E_{D}(\mathbb {Q})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> generate <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E left-parenthesis double-struck upper Q left-parenthesis StartRoot upper D EndRoot right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msqrt> <mml:mi>D</mml:mi> </mml:msqrt> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">E(\mathbb {Q}(\sqrt {D}))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We bound the proportion of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , ordered by height, for which this is not the case, showing that it is very small for typical <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The central theorem is concerned with intersections of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -Selmer groups of quadratic twists. We establish their average size in terms of a product of local densities. We additionally propose a heuristic model for these intersections, which explains our result and similar results in the literature. This heuristic predicts further results in other families.