# On the main conjectures of Iwasawa theory for certain elliptic curves with complex multiplication

dc.creator | Kezuka, Yukako | |

dc.date.accessioned | 2018-11-24T23:27:21Z | |

dc.date.available | 2017-06-19T14:02:51Z | |

dc.date.available | 2018-11-24T23:27:21Z | |

dc.date.issued | 2017-05-30 | |

dc.identifier | https://www.repository.cam.ac.uk/handle/1810/264939 | |

dc.identifier.uri | http://repository.aust.edu.ng/xmlui/handle/123456789/3967 | |

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.language | en | |

dc.publisher | University of Cambridge | |

dc.publisher | Mathematics | |

dc.subject | Iwasawa theory | |

dc.subject | Elliptic curves | |

dc.subject | Complex Multiplication | |

dc.subject | CM | |

dc.subject | Birch-Swinnerton-Dyer conjecture | |

dc.subject | BSD | |

dc.subject | main conjecture | |

dc.subject | p-adic L-function | |

dc.subject | Elliptic units | |

dc.subject | Quadratic twists | |

dc.subject | Cubic twists | |

dc.subject | L-series | |

dc.subject | Gross curve | |

dc.subject | L-value | |

dc.subject | Rubin | |

dc.title | On the main conjectures of Iwasawa theory for certain elliptic curves with complex multiplication | |

dc.type | Thesis |

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