The distribution of ℓ^{∞}-Selmer groups in degree ℓ twist families I

In this paper and its sequel, we develop a technique for finding the distribution of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l Superscript normal infinity"> <mml:semantics> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\ell ^{\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Selmer groups in degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l"> <mml:semantics> <mml:mi>ℓ</mml:mi> <mml:annotation encoding="application/x-tex">\ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula> twist families of Galois modules over number fields. Given an elliptic curve <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> over a number field satisfying certain technical conditions, this technique can be used to show that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="100"> <mml:semantics> <mml:mn>100</mml:mn> <mml:annotation encoding="application/x-tex">100%</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the quadratic twists 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> have rank at most <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding="application/x-tex">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Given a prime <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l"> <mml:semantics> <mml:mi>ℓ</mml:mi> <mml:annotation encoding="application/x-tex">\ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and a number field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> not containing <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu Subscript 2 script l"> <mml:semantics> <mml:msub> <mml:mi>μ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mi>ℓ</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mu _{2\ell }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, this method also shows that the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l Superscript normal infinity"> <mml:semantics> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\ell ^{\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-class groups in the family of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l"> <mml:semantics> <mml:mi>ℓ</mml:mi> <mml:annotation encoding="application/x-tex">\ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula> cyclic extensions of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> have a distribution consistent with the Cohen–Lenstra–Gerth heuristics. For this work, we develop the theory of the fixed point Selmer group, which serves as the base layer of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l Superscript normal infinity"> <mml:semantics> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\ell ^{\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Selmer group. This first paper gives a technique for finding the distribution of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l Superscript normal infinity"> <mml:semantics> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\ell ^{\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Selmer groups in certain families of twists where the fixed point Selmer group is stable. In the sequel paper, we will give a technique for controlling fixed point Selmer groups.