{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/264939"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/264939","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"On the main conjectures of Iwasawa theory for certain elliptic curves with complex multiplication","abstract":"The conjecture of Birch and Swinnerton-Dyer is unquestionably one of the most important open problems in number theory today. Let $E$ be an elliptic curve defined over an imaginary quadratic field $K$ contained in $\\mathbb{C}$, and suppose that $E$ has complex multiplication by the ring of integers of $K$. Let us assume the complex $L$-series $L(E/K,s)$ of $E$ over $K$ does not vanish at $s=1$. K. Rubin showed, using Iwasawa theory, that the $p$-part of Birch and Swinnerton-Dyer conjecture holds for $E$ for all prime numbers $p$ which do not divide the order of the group of roots of unity in $K$. In this thesis, we discuss extensions of this result. In Chapter $2$, we study infinite families of quadratic and cubic twists of the elliptic curve $A = X_0(27)$, so that they have complex multiplication by the ring of integers of $\\mathbb{Q}(\\sqrt{-3})$. For the family of quadratic twists, we establish a lower bound for the $2$-adic valuation of the algebraic part of the complex $L$-series at $s=1$, and, for the family of cubic twists, we establish a lower bound for the $3$-adic valuation of the algebraic part of the same $L$-value. We show that our lower bounds are precisely those predicted by Birch and Swinnerton-Dyer. In the remaining chapters, we let $K=\\mathbb{Q}(\\sqrt{-q})$, where $q$ is any prime number congruent to $7$ modulo $8$. Denote by $H$ the Hilbert class field of $K$. \\mbox{B. Gross} proved the existence of an elliptic curve $A(q)$ defined over $H$ with complex multiplication by the ring of integers of $K$ and minimal discriminant $-q^3$. We consider twists $E$ of $A(q)$ by quadratic extensions of $K$. In the case $q=7$, we have $A(q)=X_0(49)$, and Gonzalez-Aviles and Rubin proved, again using Iwasawa theory, that if $L(E/\\mathbb{Q},1)$ is nonzero then the full Birch--Swinnerton-Dyer conjecture holds for $E$. Suppose $p$ is a prime number which splits in $K$, say $p=\\mathfrak{p}\\mathfrak{p}^*$, and $E$ has good reduction at all primes of $H$ above $p$. Let $H_\\infty=HK_\\infty$, where $K_\\infty$ is the unique $\\mathbb{Z}_p$-extension of $K$ unramified outside $\\mathfrak{p}$. We establish in this thesis the main conjecture for the extension $H_\\infty/H$. Furthermore, we provide the necessary ingredients to state and prove the main conjecture for $E/H$ and $p$, and discuss its relation to the main conjecture for $H_\\infty/H$ and the $p$-part of the Birch--Swinnerton-Dyer conjecture for $E/H$.","abstract_html":"The conjecture of Birch and Swinnerton-Dyer is unquestionably one of the most important open problems in number theory today. Let $E$ be an elliptic curve defined over an imaginary quadratic field $K$ contained in $\\mathbb{C}$, and suppose that $E$ has complex multiplication by the ring of integers of $K$. Let us assume the complex $L$-series $L(E/K,s)$ of $E$ over $K$ does not vanish at $s=1$. K. Rubin showed, using Iwasawa theory, that the $p$-part of Birch and Swinnerton-Dyer conjecture holds for $E$ for all prime numbers $p$ which do not divide the order of the group of roots of unity in $K$. In this thesis, we discuss extensions of this result. In Chapter $2$, we study infinite families of quadratic and cubic twists of the elliptic curve <span class=\"etd-inline-math\">A = X<sub>0</sub>(27)</span>, so that they have complex multiplication by the ring of integers of $\\mathbb{Q}(\\sqrt{-3})$. For the family of quadratic twists, we establish a lower bound for the $2$-adic valuation of the algebraic part of the complex $L$-series at $s=1$, and, for the family of cubic twists, we establish a lower bound for the $3$-adic valuation of the algebraic part of the same $L$-value. We show that our lower bounds are precisely those predicted by Birch and Swinnerton-Dyer. In the remaining chapters, we let $K=\\mathbb{Q}(\\sqrt{-q})$, where $q$ is any prime number congruent to $7$ modulo $8$. Denote by $H$ the Hilbert class field of $K$. \\mbox{B. Gross} proved the existence of an elliptic curve $A(q)$ defined over $H$ with complex multiplication by the ring of integers of $K$ and minimal discriminant <span class=\"etd-inline-math\">-q<sup>3</sup></span>. We consider twists $E$ of $A(q)$ by quadratic extensions of $K$. In the case $q=7$, we have <span class=\"etd-inline-math\">A(q)=X<sub>0</sub>(49)</span>, and Gonzalez-Aviles and Rubin proved, again using Iwasawa theory, that if $L(E/\\mathbb{Q},1)$ is nonzero then the full Birch--Swinnerton-Dyer conjecture holds for $E$. Suppose $p$ is a prime number which splits in $K$, say <span class=\"etd-inline-math\">p=\\mathfrak{p}\\mathfrak{p}<sup>*</sup></span>, and $E$ has good reduction at all primes of $H$ above $p$. Let <span class=\"etd-inline-math\">H<sub>\\</sub>infty=HK<sub>\\</sub>infty</span>, where <span class=\"etd-inline-math\">K<sub>\\</sub>infty</span> is the unique <span class=\"etd-inline-math\">\\mathbb{Z}<sub>p</sub></span>-extension of $K$ unramified outside $\\mathfrak{p}$. We establish in this thesis the main conjecture for the extension <span class=\"etd-inline-math\">H<sub>\\</sub>infty/H</span>. Furthermore, we provide the necessary ingredients to state and prove the main conjecture for $E/H$ and $p$, and discuss its relation to the main conjecture for <span class=\"etd-inline-math\">H<sub>\\</sub>infty/H</span> and the $p$-part of the Birch--Swinnerton-Dyer conjecture for $E/H$.","abstract_has_math":true,"creators":["Kezuka, Yukako"],"institution":"University of Cambridge","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Coates, John Henry"],"committee_chairs":[],"committee_members":[],"year":2017,"date_issued":"2017-05-30","date_published":"2017-05-30","updated_at":"2026-07-22T22:24:14Z","subjects":["Iwasawa theory","Elliptic curves","Complex Multiplication","CM","Birch-Swinnerton-Dyer conjecture","BSD","main conjecture","p-adic L-function","Elliptic units","Quadratic twists","Cubic twists","L-series","Gross curve","L-value","Rubin"],"languages":["en"],"rights":[],"rights_urls":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/26dd4726-c3a5-4c55-a407-d90de98c25fb/download","https://www.rioxx.net/licenses/all-rights-reserved/"],"identifier_entries":[]},"links":{"outbound_url":"https://doi.org/10.17863/CAM.10705","outbound_label":"DOI","outbound_source":"dc:identifier.doi"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Coates, John Henry"]},{"key":"dc:creator","label":"Author","values":["Kezuka, Yukako"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2017-05-30"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Cambridge"]},{"key":"dc:relation.isreferencedby.uri","label":"Dc Relation Isreferencedby URI","values":["https://www.repository.cam.ac.uk/handle/1810/264939"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Iwasawa theory","Elliptic curves","Complex Multiplication","CM","Birch-Swinnerton-Dyer conjecture","BSD","main conjecture","p-adic L-function","Elliptic units","Quadratic twists","Cubic twists","L-series","Gross curve","L-value","Rubin"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/26dd4726-c3a5-4c55-a407-d90de98c25fb/download","https://www.rioxx.net/licenses/all-rights-reserved/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["10.17863/CAM.10705"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/d3d613e1-6d73-4177-b724-9148d9d83073/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The conjecture of Birch and Swinnerton-Dyer is unquestionably one of the most important open problems in number theory today. Let $E$ be an elliptic curve defined over an imaginary quadratic field $K$ contained in $\\mathbb{C}$, and suppose that $E$ has complex multiplication by the ring of integers of $K$. Let us assume the complex $L$-series $L(E/K,s)$ of $E$ over $K$ does not vanish at $s=1$. K. Rubin showed, using Iwasawa theory, that the $p$-part of Birch and Swinnerton-Dyer conjecture holds for $E$ for all prime numbers $p$ which do not divide the order of the group of roots of unity in $K$. In this thesis, we discuss extensions of this result. In Chapter $2$, we study infinite families of quadratic and cubic twists of the elliptic curve $A = X_0(27)$, so that they have complex multiplication by the ring of integers of $\\mathbb{Q}(\\sqrt{-3})$. For the family of quadratic twists, we establish a lower bound for the $2$-adic valuation of the algebraic part of the complex $L$-series at $s=1$, and, for the family of cubic twists, we establish a lower bound for the $3$-adic valuation of the algebraic part of the same $L$-value. We show that our lower bounds are precisely those predicted by Birch and Swinnerton-Dyer. In the remaining chapters, we let $K=\\mathbb{Q}(\\sqrt{-q})$, where $q$ is any prime number congruent to $7$ modulo $8$. Denote by $H$ the Hilbert class field of $K$. \\mbox{B. Gross} proved the existence of an elliptic curve $A(q)$ defined over $H$ with complex multiplication by the ring of integers of $K$ and minimal discriminant $-q^3$. We consider twists $E$ of $A(q)$ by quadratic extensions of $K$. In the case $q=7$, we have $A(q)=X_0(49)$, and Gonzalez-Aviles and Rubin proved, again using Iwasawa theory, that if $L(E/\\mathbb{Q},1)$ is nonzero then the full Birch--Swinnerton-Dyer conjecture holds for $E$. Suppose $p$ is a prime number which splits in $K$, say $p=\\mathfrak{p}\\mathfrak{p}^*$, and $E$ has good reduction at all primes of $H$ above $p$. Let $H_\\infty=HK_\\infty$, where $K_\\infty$ is the unique $\\mathbb{Z}_p$-extension of $K$ unramified outside $\\mathfrak{p}$. We establish in this thesis the main conjecture for the extension $H_\\infty/H$. Furthermore, we provide the necessary ingredients to state and prove the main conjecture for $E/H$ and $p$, and discuss its relation to the main conjecture for $H_\\infty/H$ and the $p$-part of the Birch--Swinnerton-Dyer conjecture for $E/H$."]},{"key":"dc:format.checksum.md5","label":"Dc Format Checksum Md5","values":["afa0f3a92c3384d5fa583073ea822027","87eda9de84448d1f82354d60eee3eb5f"]},{"key":"dc:title","label":"Title","values":["On the main conjectures of Iwasawa theory for certain elliptic curves with complex multiplication"]}]}],"canonical_facts":{"dc:contributor.advisor":["Coates, John Henry"],"dc:creator":["Kezuka, Yukako"],"dc:date.issued":["2017-05-30"],"dc:description.abstract":["The conjecture of Birch and Swinnerton-Dyer is unquestionably one of the most important open problems in number theory today. Let $E$ be an elliptic curve defined over an imaginary quadratic field $K$ contained in $\\mathbb{C}$, and suppose that $E$ has complex multiplication by the ring of integers of $K$. Let us assume the complex $L$-series $L(E/K,s)$ of $E$ over $K$ does not vanish at $s=1$. K. Rubin showed, using Iwasawa theory, that the $p$-part of Birch and Swinnerton-Dyer conjecture holds for $E$ for all prime numbers $p$ which do not divide the order of the group of roots of unity in $K$. In this thesis, we discuss extensions of this result. In Chapter $2$, we study infinite families of quadratic and cubic twists of the elliptic curve $A = X_0(27)$, so that they have complex multiplication by the ring of integers of $\\mathbb{Q}(\\sqrt{-3})$. For the family of quadratic twists, we establish a lower bound for the $2$-adic valuation of the algebraic part of the complex $L$-series at $s=1$, and, for the family of cubic twists, we establish a lower bound for the $3$-adic valuation of the algebraic part of the same $L$-value. We show that our lower bounds are precisely those predicted by Birch and Swinnerton-Dyer. In the remaining chapters, we let $K=\\mathbb{Q}(\\sqrt{-q})$, where $q$ is any prime number congruent to $7$ modulo $8$. Denote by $H$ the Hilbert class field of $K$. \\mbox{B. Gross} proved the existence of an elliptic curve $A(q)$ defined over $H$ with complex multiplication by the ring of integers of $K$ and minimal discriminant $-q^3$. We consider twists $E$ of $A(q)$ by quadratic extensions of $K$. In the case $q=7$, we have $A(q)=X_0(49)$, and Gonzalez-Aviles and Rubin proved, again using Iwasawa theory, that if $L(E/\\mathbb{Q},1)$ is nonzero then the full Birch--Swinnerton-Dyer conjecture holds for $E$. Suppose $p$ is a prime number which splits in $K$, say $p=\\mathfrak{p}\\mathfrak{p}^*$, and $E$ has good reduction at all primes of $H$ above $p$. Let $H_\\infty=HK_\\infty$, where $K_\\infty$ is the unique $\\mathbb{Z}_p$-extension of $K$ unramified outside $\\mathfrak{p}$. We establish in this thesis the main conjecture for the extension $H_\\infty/H$. Furthermore, we provide the necessary ingredients to state and prove the main conjecture for $E/H$ and $p$, and discuss its relation to the main conjecture for $H_\\infty/H$ and the $p$-part of the Birch--Swinnerton-Dyer conjecture for $E/H$."],"dc:format.checksum.md5":["afa0f3a92c3384d5fa583073ea822027","87eda9de84448d1f82354d60eee3eb5f"],"dc:identifier.doi":["10.17863/CAM.10705"],"dc:identifier.uri":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/d3d613e1-6d73-4177-b724-9148d9d83073/download"],"dc:language":["en"],"dc:publisher.institution":["University of Cambridge"],"dc:relation.isreferencedby.uri":["https://www.repository.cam.ac.uk/handle/1810/264939"],"dc:rights":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/26dd4726-c3a5-4c55-a407-d90de98c25fb/download","https://www.rioxx.net/licenses/all-rights-reserved/"],"dc:subject":["Iwasawa theory","Elliptic curves","Complex Multiplication","CM","Birch-Swinnerton-Dyer conjecture","BSD","main conjecture","p-adic L-function","Elliptic units","Quadratic twists","Cubic twists","L-series","Gross curve","L-value","Rubin"],"dc:title":["On the main conjectures of Iwasawa theory for certain elliptic curves with complex multiplication"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["Doctoral"],"dc:type.qualificationname":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-22T22:24:14Z"}