## Abstract

. Fix an elliptic curve E over a number field F and an integer п which is a power of 3. We study the growth of the Mordell—Weil rank of E after base change to the fields (Formula Presented). If E admits a 3-isogeny, then we show that the average "new rank" of E over K_{d}, appropriately defined, is bounded as the height of d goes to infinity. When п = 3, we moreover show that for many elliptic curves E/Q, there are no new points on E over (Formula Presented), for a positive proportion of integers d. This is a horizontal analogue of a well-known result of Cornut and Vatsal [Nontriviality of Rankin-Selberg L-functions and CM points, L-functions and Galois representations, vol. 320, Cambridge Univ. Press, Cambridge, 2007, pp. 121—186]. As a corollary, we show that Hilbert's tenth problem has a negative solution over a positive proportion of pure sextic fields (Formula Presented). The proofs combine our recent work on ranks of abelian varieties in су-clotomic twist families with a technique we call the "correlation trick", which applies in a more general context where one is trying to show simultaneous vanishing of multiple Selmer groups. We also apply this technique to families of twists of Prym surfaces, which leads to bounds on the number of rational points in sextic twist families of bielliptic genus 3 curves.

Original language | American English |
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Pages (from-to) | 482-506 |

Number of pages | 25 |

Journal | Transactions of the American Mathematical Society Series B |

Volume | 10 |

DOIs | |

State | Published - 2023 |

### Bibliographical note

Publisher Copyright:© 2023 by the author(s).