Lattices and Rational Points

Abstract

Abstract:In this article, we show how to use the first and second Minkowski Theorems and some Diophantine geometry to bound explicitly the height of the points of rank N − 1 on transverse curves in E N , where E is an elliptic curve without Complex Multiplication (CM). We then apply our result to give a method for finding the rational points on such curves, when E has Q-rank ≤ N − 1. We also give some explicit examples. This result generalises from rank 1 to rank N − 1 previous results of S. Checcoli, F. Veneziano and the author.

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

“…Central to this purpose is the use of the first Minkowski theorem instead of other easier but less sharp approximations. This proof generalizes the proof in [CVV19] from rank one to higher rank and the proof of [Via17] from Z-lattices to O-lattices, with O an order in the ring of integers of an imaginary quadratic number field. This completes all cases of the explicit Mordell Conjecture that can be covered with our method and opens a wide range of examples of curves suitable for determining the k-rational points.…”

“…In that paper we tested the possibility of producing an explicit and even implementable method for finding the rational points on some new families of algebraic curves. In [Via18] and [Via17] Viada extended the previous methods of [CVV17] and [CVV19], obtaining partial and less sharp results that are however too large to be implemented.…”