2017/07/16 by Peter Bruin, Bruin, Peter, Andrea Ferraguti +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.1707.04861
17 pages; comments are welcome!
arxiv created 2017/07/16 · openalex publication_date 2017/07/16 · arxiv updated 2017/07/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let E be a \mathbb Q-curve without complex multiplication. We address the problem of deciding whether E is geometrically isomorphic to a strongly modular \mathbb Q-curve. We show that the question has a positive answer if and only if E has a model that is completely defined over an abelian number field. Next, if E is completely defined over a quadratic or biquadratic number field L, we classify all strongly modular twists of E over L in terms of the arithmetic of L. Moreover, we show how to determine which of these twists come, up to isogeny, from a subfield of L.