Explicit height bounds for $K$-rational points on transverse curves in powers of elliptic curves

Abstract

Let C be an algebraic curve embedded transversally in a power E N of an elliptic curve E. In this article we produce a good explicit bound for the height of all the algebraic points on C contained in the union of all proper algebraic subgroups of E N . The method gives a totally explicit version of the Manin-Dam'janenko Theorem in the elliptic case and it is a generalisation of previous results only proved when E does not have Complex Multiplication.

Recent developments of the Uniform Mordell-Lang Conjecture

“…Little is known for this conjecture. In spirit of the Manin-Demjanenko method [Ser13, §5.2], Checcoli, Veneziano, and Viada [CVV17,VV20] have some results on this. There are also p-adic approaches (Chabauty-Coleman-Kim, Lawrence-Venkatesh) to this question, for which we refer to the survey [BBB + 21].…”