Elliptic Curves with Long Arithmetic Progressions Have Large Rank

Abstract

For any family of elliptic curves over the rational numbers with fixed j-invariant, we prove that the existence of a long sequence of rational points whose x-coordinates form a non-trivial arithmetic progression implies that the Mordell-Weil rank is large, and similarly for y-coordinates. We give applications related to uniform boundedness of ranks, conjectures by Bremner and Mohanty, and arithmetic statistics on elliptic curves. Our approach involves Nevanlinna theory as well as Rémond's quantitative extension of results of Faltings.

Mestre's Finite Field Method for Searching Elliptic Curves with High Ranks

“…Recent research in rank studies has tackled three key areas: calculating ranks for specific families of curves [15][16][17][18][19], exploring how rank behaves for curves constructed from special number sequences [20][21][22][23][24][25][26][27][28] and analyzing rank distributions within families and across field extensions [29][30][31]. Dujella [32] provided an enumeration of the strategies for generating high-rank Diaphontine elliptic curves.…”