{"title":"Two Space-Filling Curves","authors":"","doi":"10.2307/j.ctvfrxr22.10","DOIUrl":"https://doi.org/10.2307/j.ctvfrxr22.10","url":null,"abstract":"","PeriodicalId":416024,"journal":{"name":"Curves for the Mathematically Curious","volume":"94 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134336074","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Weierstrass Curve","authors":"","doi":"10.2307/j.ctvfrxr22.6","DOIUrl":"https://doi.org/10.2307/j.ctvfrxr22.6","url":null,"abstract":"","PeriodicalId":416024,"journal":{"name":"Curves for the Mathematically Curious","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115339776","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Perhaps the Most Important Curve of All","authors":"","doi":"10.2307/j.ctvfrxr22.15","DOIUrl":"https://doi.org/10.2307/j.ctvfrxr22.15","url":null,"abstract":"","PeriodicalId":416024,"journal":{"name":"Curves for the Mathematically Curious","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128921400","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Elliptic Curves","authors":"Samuel S. Wagstaff","doi":"10.1201/9781315275765-12","DOIUrl":"https://doi.org/10.1201/9781315275765-12","url":null,"abstract":"The main focus of this paper is the study of elliptic curves, non-singular projective curves of genus 1. Under a geometric operation, the rational points E(Q) of an elliptic curve E form a group, which is a finitely-generated abelian group by Mordell’s theorem. Thus, this group can be expressed as the finite direct sum of copies of Z and finite cyclic groups. The number of finite copies of Z is called the rank of E(Q). From John Tate and Joseph Silverman [ST92], we have a formula to compute the rank of curves of the form E : y2 = x3 + ax2 + bx. In this thesis, we generalize this formula, using a purely group theoretic approach, and utilize this generalization to find the rank of curves of the form E : y2 = x3 + c. To do this, we review a few well-known homomorphisms on the curve E : y2 = x3 + ax2 + bx as in Tate and Silverman’s Elliptic Curves [ST92], and study analogous homomorphisms on E : y2 = x3 + c and relevant facts.","PeriodicalId":416024,"journal":{"name":"Curves for the Mathematically Curious","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116377979","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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