{"id":{"repo_id":"columbia-diss","oai_identifier":"oai:academiccommons.columbia.edu:10.7916/D8CZ3FD0"},"canonical_url":"https://search.dev.ndltd.org/etd/columbia-diss/oai:academiccommons.columbia.edu:10.7916/D8CZ3FD0","repository":{"repo_id":"columbia-diss","name":"Columbia University","base_url":"https://academiccommons.columbia.edu/oai"},"display":{"title":"p-adic Heights of Heegner points on Shimura curves","abstract":"Let f be a primitive Hilbert modular form of weight 2 and level N for the totally real field F, and let p be an odd rational prime such that f is ordinary at all primes dividing p. When E is a CM extension of F of relative discriminant prime to Np, we give an explicit construction of the p-adic Rankin-Selberg L-function L_p(f_E,-) and prove that when the sign of its functional equation is -1, its central derivative is given by the p-adic height of a Heegner point on the abelian variety A associated to f. This p-adic Gross-Zagier formula generalises the result obtained by Perrin-Riou when F=Q and N satisfies the so-called Heegner condition. We deduce applications to both the p-adic and the classical Birch and Swinnerton-Dyer conjectures for A.","abstract_html":"Let f be a primitive Hilbert modular form of weight 2 and level N for the totally real field F, and let p be an odd rational prime such that f is ordinary at all primes dividing p. When E is a CM extension of F of relative discriminant prime to Np, we give an explicit construction of the p-adic Rankin-Selberg L-function L_p(f_E,-) and prove that when the sign of its functional equation is -1, its central derivative is given by the p-adic height of a Heegner point on the abelian variety A associated to f. This p-adic Gross-Zagier formula generalises the result obtained by Perrin-Riou when F=Q and N satisfies the so-called Heegner condition. We deduce applications to both the p-adic and the classical Birch and Swinnerton-Dyer conjectures for A.","abstract_has_math":false,"creators":["Disegni, Daniel"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2013,"date_issued":"2013","date_published":"2013","updated_at":"2026-07-24T01:44:27Z","subjects":["Mathematics","Modular curves","Number theory"],"languages":["English"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://doi.org/10.7916/D8CZ3FD0","outbound_label":"DOI","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Disegni, Daniel"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2013"]},{"key":"dc:type","label":"Dc Type","values":["Theses"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics","Modular curves","Number theory"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["English"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://doi.org/10.7916/D8CZ3FD0"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Let f be a primitive Hilbert modular form of weight 2 and level N for the totally real field F, and let p be an odd rational prime such that f is ordinary at all primes dividing p. When E is a CM extension of F of relative discriminant prime to Np, we give an explicit construction of the p-adic Rankin-Selberg L-function L_p(f_E,-) and prove that when the sign of its functional equation is -1, its central derivative is given by the p-adic height of a Heegner point on the abelian variety A associated to f. This p-adic Gross-Zagier formula generalises the result obtained by Perrin-Riou when F=Q and N satisfies the so-called Heegner condition. We deduce applications to both the p-adic and the classical Birch and Swinnerton-Dyer conjectures for A."]},{"key":"dc:title","label":"Title","values":["p-adic Heights of Heegner points on Shimura curves"]}]}],"canonical_facts":{"dc:creator":["Disegni, Daniel"],"dc:date":["2013"],"dc:description":["Let f be a primitive Hilbert modular form of weight 2 and level N for the totally real field F, and let p be an odd rational prime such that f is ordinary at all primes dividing p. When E is a CM extension of F of relative discriminant prime to Np, we give an explicit construction of the p-adic Rankin-Selberg L-function L_p(f_E,-) and prove that when the sign of its functional equation is -1, its central derivative is given by the p-adic height of a Heegner point on the abelian variety A associated to f. This p-adic Gross-Zagier formula generalises the result obtained by Perrin-Riou when F=Q and N satisfies the so-called Heegner condition. We deduce applications to both the p-adic and the classical Birch and Swinnerton-Dyer conjectures for A."],"dc:identifier":["https://doi.org/10.7916/D8CZ3FD0"],"dc:language":["English"],"dc:subject":["Mathematics","Modular curves","Number theory"],"dc:title":["p-adic Heights of Heegner points on Shimura curves"],"dc:type":["Theses"]},"updated_at":"2026-07-24T01:44:27Z"}