{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/104589"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/104589","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Counting elliptic curves of bounded Faltings height","abstract":"Because many invariants and properties of elliptic curves are difficult to understand directly, the study of arithmetic statistics instead looks at what happens \"on average\", using heights to make this notion rigorous. Previous work has primarily used the naive height, which can be calculated easily but is not defined intrinsically. We give an asymptotic formula for the number of elliptic curves over Q with bounded Faltings height. Silverman [34] has shown that the Faltings height for elliptic curves over number fields can be expressed in terms of the minimal discriminant and period of the elliptic curve. We use this to recast the problem as one of counting lattice points in an unbounded region in R2 defined by transcendental equations, and understand this region well enough to give a formula for the number of these lattice points.","abstract_html":"Because many invariants and properties of elliptic curves are difficult to understand directly, the study of arithmetic statistics instead looks at what happens "on average", using heights to make this notion rigorous. Previous work has primarily used the naive height, which can be calculated easily but is not defined intrinsically. We give an asymptotic formula for the number of elliptic curves over Q with bounded Faltings height. Silverman [34] has shown that the Faltings height for elliptic curves over number fields can be expressed in terms of the minimal discriminant and period of the elliptic curve. We use this to recast the problem as one of counting lattice points in an unbounded region in R2 defined by transcendental equations, and understand this region well enough to give a formula for the number of these lattice points.","abstract_has_math":false,"creators":["Hortsch, Ruthi"],"institution":"Massachusetts Institute of Technology","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. Department of Mathematics.","school":null,"contributors":[],"advisors":["Bjorn Poonen."],"committee_chairs":[],"committee_members":[],"year":2016,"date_issued":"2016","date_published":"2016","updated_at":"2026-07-22T22:21:26Z","subjects":["Mathematics."],"languages":["eng"],"rights":["M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. 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D., Massachusetts Institute of Technology, Department of Mathematics, 2016.","Cataloged from PDF version of thesis.","Includes bibliographical references (pages 47-50)."],"dc:description.abstract":["Because many invariants and properties of elliptic curves are difficult to understand directly, the study of arithmetic statistics instead looks at what happens \"on average\", using heights to make this notion rigorous. Previous work has primarily used the naive height, which can be calculated easily but is not defined intrinsically. We give an asymptotic formula for the number of elliptic curves over Q with bounded Faltings height. Silverman [34] has shown that the Faltings height for elliptic curves over number fields can be expressed in terms of the minimal discriminant and period of the elliptic curve. We use this to recast the problem as one of counting lattice points in an unbounded region in R2 defined by transcendental equations, and understand this region well enough to give a formula for the number of these lattice points."],"dc:description.degree":["Ph. D."],"dc:identifier.uri":["http://hdl.handle.net/1721.1/104589"],"dc:language.iso":["eng"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission."],"dc:rights.uri":["http://dspace.mit.edu/handle/1721.1/7582"],"dc:subject":["Mathematics."],"dc:title":["Counting elliptic curves of bounded Faltings height"],"dc:type":["Thesis"]},"updated_at":"2026-07-22T22:21:26Z"}