2026/01/09 by Luca Ferrigno
#math.NT #math.AG
Let S be a smooth irreducible curve over ℚ, and let A → S be an abelian scheme with a curve C ⊂ A, both defined over ℚ. In 2020, Barroero and Capuano proved that if C is not contained in a proper subgroup scheme, then the intersection of C with the union of the flat subgroup schemes of A of codimension at least 2 is finite. In this article, we continue to study this problem by considering the intersections with the algebraic subgroups of the CM fibers, generalizing a previous result of Barroero for fibered powers of elliptic schemes. A key ingredient of the proof is an explicit control of canonical heights under endomorphisms: for an abelian variety A/ℚ, an ample symmetric divisor D, and f ∈ End(A), we bound explicitly \widehathA, D(f(P)) in terms of \widehathA, D(P) by determining the values of λ∈ ℝ for which the divisors λD - f^* D and f^* D - λD are ample.