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

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 = X₀(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 -q³. We consider twists $E$ of $A(q)$ by quadratic extensions of $K$. In the case $q=7$, we have A(q)=X₀(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 $E/H$.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2017

Author dc:creator

• Kezuka, Yukako

Advisor dc:contributor.advisor

• Coates, John Henry

• 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

Licence

• https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/26dd4726-c3a5-4c55-a407-d90de98c25fb/download
• https://www.rioxx.net/licenses/all-rights-reserved/

Language dc:language

en

DOI dc:identifier.doi

https://doi.org/10.17863/CAM.10705

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/264939