2026/07/01 by D. Masser, U. Zannier
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #Polynomial and algebraic computation
paper · doi:10.1112/jlms.70625
Abstract In a recent paper, we found two counterexamples to a specialization conjecture of James Davenport involving integrals on algebraic curves. We showed that every such elliptic counterexample must involve complex multiplication; ours corresponded to the fields and . Here, we prove that every imaginary also turns up. This therefore becomes analogous to the situation with relative Manin‐Mumford discovered by Daniel Bertrand, even though our underlying algebraic groups are rather different from his. We also give a complete description of all elliptic counterexamples (our paper could handle only one at a time). A consequence is that there is a counterexample on an algebraic curve if and only if its Jacobian contains an elliptic curve with complex multiplication.