2022/06/30 by Francesco Ballini, Ballini, Francesco
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Polynomial and algebraic computation #Commutative Algebra and Its Applications
paper · pdf · doi:10.48550/arxiv.2207.00049
Let A be an abelian variety defined over a number field K. We say that a point P ∈ A(ℚ) is primitive if there is no Q ∈ A(ℚ) defined on the field of definition of P over K such that [N]Q=P for some positive integer N ≥ 2. For any primitive point P ∈ A(ℚ), positive integer N and point Q ∈ A(ℚ) such that [N]Q=P, we prove an effective lower bound on the degree of the field of definition of Q over K of the form Nδ that depends only on A,K and the degree of the field of definition of P over K. The proof is based on the estimates of the degree of torsion points by Masser. We combine this result with a uniform version of Manin-Mumford to prove an effective Unlikely Intersections-type result: if P ∈ A(ℚ) is primitive, defined over a field of degree d over K, and X is a subvariety of A, then X ∩ [N]-1P is contained in the weakly special part of X, provided N is bigger than a suitable power of d. As an application, we study an inverse elliptic Fermat equation, analogous to a modular Fermat equation treated by Pila.