TY - JOUR

T1 - RANK GROWTH OF ELLIPTIC CURVES OVER N -TH ROOT EXTENSIONS

AU - Shnidman, Ari

AU - Weiss, Ariel

N1 - Funding Information:
Received by the editors January 23, 2022, and, in revised form, November 22, 2022, and December 27, 2022. 2020 Mathematics Subject Classification. Primary 11G05, 14G05, 14K05, 11S25. The first author was supported by the Israel Science Foundation (grant No. 2301/20). The second author was supported by an Emily Erskine Endowment Fund postdoctoral fellowship at the Hebrew University of Jerusalem, by the Israel Science Foundation (grant No. 1963/20) and by the US-Israel Binational Science Foundation (grant No. 2018250).
Publisher Copyright:
© 2023 by the author(s).

PY - 2023/1/1

Y1 - 2023/1/1

N2 - . 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 Kd, 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.

AB - . 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 Kd, 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.

UR - http://www.scopus.com/inward/record.url?scp=85159717103&partnerID=8YFLogxK

U2 - 10.1090/btran/149

DO - 10.1090/btran/149

M3 - Article

AN - SCOPUS:85159717103

SN - 2330-0000

VL - 10

SP - 482

EP - 506

JO - Transactions of the American Mathematical Society Series B

JF - Transactions of the American Mathematical Society Series B

ER -