Analytic ranks of elliptic curves over number fields

Let E E be an elliptic curve over alttext="double-struck upper Q"> Q encoding="application/x-tex">\mathbb {Q} . Then, we show that the average analytic rank of cyclic extensions degree alttext="l"> l encoding="application/x-tex">l with a prime not equal to alttext="2"> 2 encoding="application/x-tex">2 , is at most alttext="2 plus r Subscript double-struck Q Baseline left-parenthesis E right-parenthesis"> + r ( stretchy="false">) encoding="application/x-tex">2+r_\mathbb {Q}(E) where alttext="r encoding="application/x-tex">r_\mathbb This bound independent Using recent result Bhargava, Taniguchi and Thorne [Improved error estimates for Davenport–Heilbronn theorems, arxiv.org/abs/2107.12819, 2021], obtain non-trivial on S 3"> S 3 encoding="application/x-tex">S_3 -fields.