Title: Unlikely intersections in families of abelian varieties and the polynomial Pell equation
Abstract: Proceedings of the London Mathematical SocietyVolume 120, Issue 2 p. 192-219 Research Article Unlikely intersections in families of abelian varieties and the polynomial Pell equation F. Barroero, F. Barroero [email protected] Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, Largo San Murialdo 1, 00146 Roma, ItalySearch for more papers by this authorL. Capuano, L. Capuano [email protected] Dipartimento di Scienze Matematiche “Giuseppe Luigi Lagrange”, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, ItalySearch for more papers by this author F. Barroero, F. Barroero [email protected] Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, Largo San Murialdo 1, 00146 Roma, ItalySearch for more papers by this authorL. Capuano, L. Capuano [email protected] Dipartimento di Scienze Matematiche “Giuseppe Luigi Lagrange”, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, ItalySearch for more papers by this author First published: 02 September 2019 https://doi.org/10.1112/plms.12289Citations: 3 This work was supported by the European Research Council [267273], the Engineering and Physical Sciences Research Council [EP/N007956/1 to F. B. and EP/N008359/1 to L. C.], the Istituto Nazionale di Alta Matematica [Borsa Ing. G. Schirillo to L. C.] and the Swiss National Science Foundation [165525 to F. B.]. Read the full textAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Let S be a smooth irreducible curve defined over a number field k and consider an abelian scheme A over S and a curve C inside A, both defined over k. In previous works, we proved that, when A is a fibred product of elliptic schemes, if C is not contained in a proper subgroup scheme of A, then it contains at most finitely many points that belong to a flat subgroup scheme of codimension at least 2. In this article, we continue our investigation and settle the crucial case of powers of simple abelian schemes of relative dimension g ⩾ 2 . This, combined with the above-mentioned result and work by Habegger and Pila, gives the statement for general abelian schemes which has applications in the study of solvability of almost-Pell equations in polynomials. Citing Literature Volume120, Issue2February 2020Pages 192-219 RelatedInformation