Effective height bounds via Mordell Weil lattice symmetries

Abstract

We obtain explicit, computable upper bounds for the Neron-Tate height of rational points on curves of genus at least two over number fields. The bounds use automorphisms acting on the Mordell-Weil lattice of the Jacobian. We prove an averaged spectral-gap criterion that replaces the 'large enough automorphism group' requirement of prior work. As a special case, when some automorphism acts trivially on V the method gives the sharper bound. In rank 2 we provide a practical Bravais-lattice test, using the height Gram matrix, to detect when this situation occurs. We illustrate the method on a genus-2, rank-2 curve where the earlier large-automorphisms hypothesis does not apply.