Ranks of elliptic curves in cyclic sextic extensions of $\mathbb{Q}$

Abstract

For an elliptic curve $E/\mathbb{Q}$ we show that there are infinitely many cyclic sextic extensions $K/\mathbb{Q}$ such that the Mordell-Weil group $E(K)$ has rank greater than the subgroup of $E(K)$ generated by all the $E(F)$ for the proper subfields $F \subset K$. For certain curves $E/\mathbb{Q}$ we show that the number of such fields $K$ of conductor less than $X$ is $\gg\sqrt X$.

Figures (2)

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Theorems & Definitions (33)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• Remark 2.5
• proof
• Remark 2.6