New unlikely intersections on elliptic surfaces

2025/08/08 by Douglas Ulmer, José Felipe Voloch, Ulmer, Douglas +1
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2508.06680

openalex publication_date 2025/08/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Consider a Jacobian elliptic surface E → C with a section P of infinite order. Previous work of the first author and Urzúa over the complex numbers gives a bound on the number of tangencies between P and a torsion section of E (an ``unlikely intersection''), and more precisely, an exact formula for the weighted number of tangencies between P and elements of the ``Betti foliation''. This work used analytic techniques that apparently do not generalize to positive characteristic. In this paper, we extend their work to characteristic p, and we develop a second approach to tangency properties of algebraic curves on a complex elliptic surface, yielding a new family of unlikely intersections with a strong connection to a famous homomorphism of Manin. We also correct inaccuracies in the literature about this homomorphism.

Citations

• On the torsion values for sections of an elliptic scheme

Related

• Torsion of elliptic curves and unlikely intersections
• Transversality of sections on elliptic surfaces with applications to elliptic divisibility sequences and geography of surfaces
• Bounding tangencies of sections on elliptic surfaces
• Unlikely and just likely intersections for high dimensional families of elliptic curves
• Multiplicities of the Betti map associated to a section of an elliptic surface from a differential-geometric perspective