Ranks of elliptic curves over cyclic cubic, quartic, and sextic extensions

Neeraj Kashyap

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arXiv · 1401.8179

Ranks of elliptic curves over cyclic cubic, quartic, and sextic extensions

Neeraj Kashyap

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Abstract

For a given group $G$ and an elliptic curve $E$ defined over a number field $K$, I discuss the problem of finding $G$-extensions of $K$ over which $E$ gains rank. I prove the following theorem, extending a result of Fearnley, Kisilevsky, and Kuwata: Let $n = 3,4,$ or $6$. If $K$ contains its $n^{th}$-roots of unity then, for any elliptic curve $E$ over $K$, there are infinitely many $\mathbb{Z}/n\mathbb{Z}$-extensions of $K$ over which $E$ gains rank.

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Neeraj Kashyap. 2014-01-31. Ranks of elliptic curves over cyclic cubic, quartic, and sextic extensions. https://arxiv.org/abs/1401.8179

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