2004/08/11 by Brian Conrad, B. Conrad, Keith Conrad +6 · 1 citation
Mathematics · #11G05 #11G40 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #math.AG #math.NT #msc:11G05 #msc:11G40
paper · pdf · doi:10.48550/arxiv.math/0408153
40 pages; last version; to appear in Adv. Math
openalex publication_date 2004/08/11 · arxiv created 2005/06/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a global field K and an elliptic curve Eeta over K(T), Silverman's specialization theorem implies that rank(Eeta(K(T))) <= rank(Et(K)) for all but finitely many t in P1(K). If this inequality is strict for all but finitely many t, the elliptic curve Eeta is said to have elevated rank. All known examples of elevated rank for K=Q rest on the parity conjecture for elliptic curves over Q, and the examples are all isotrivial. Some additional standard conjectures over Q imply that there does not exist a non-isotrivial elliptic curve over Q(T) with elevated rank. In positive characteristic, an analogue of one of these additional conjectures is false. Inspired by this, for the rational function field K = kappa(u) over any finite field kappa with odd characteristic, we construct an explicit 2-parameter family Ec,d of non-isotrivial elliptic curves over K(T) (depending on arbitrary c, d in kappa^*) such that, under the parity conjecture, each Ec,d has elevated rank.