An Explicit Manin-Dem’janenko Theorem in Elliptic Curves

Abstract

Abstract. Let C be a curve of genus at least embedded in E × ⋅ ⋅ ⋅ × E N where the E i are elliptic curves for i = , . . . , N. In this article we give an explicit sharp bound for the Néron-Tate height of the points of C contained in the union of all algebraic subgroups of dimension < max(r C − t C , t C ) where t C , respectively r C , is the minimal dimension of a translate, respectively of a torsion variety, containing C.As a corollary, we give an explicit bound for the height of the rational points of special curves, proving new cases of the explicit Mordell Conjecture and in particular making explicit (and slightly more general in the CM case) the Manin-Dem'janenko method in products of elliptic curves.

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

“…This bound is sharp and it improves by a factor 6N 12 N −1 the bound in [Via18]. If N = 2 we recover (for curves with CM as well) the same bound as in [CVV19].…”

“…In order to give a general bound for the height of H + P we use an argument based on linear algebra, and some bounds on heights from Subsection 2. We prove the following proposition, which generalises [CVV19], Proposition 5.1 and improves on [Via18], Proposition 4.1: Proposition 3.4. Let P be a point in E N .…”

“…Then we have the following special case of [BG06], Theorem C.2.11. See [Via18] for the deduction of Theorem 2.4 from [BG06], Theorem C.2.11 and C.2.18.…”

“…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.…”

Lattices and Rational Points

“…We also give a non-density result for the points of rank one on a weak-transverse variety of E N . In [19], the author extends the method for curves in E N , where E has CM. Unfortunately, these bounds are much too big to be used to find the rational points on any curve.…”