Vijest
NEW DOCTORAL DEGREES Elliptic curves of large rank over quadratic fields
Mirela Jukić Bokun
; Department of Mathematics, University of Osijek, Osijek, Croatia
Sažetak
In this thesis, we study the construction of elliptic curves of positive and relatively large rank with a fixed torsion group over quadratic fields.
If $K$ is a quadratic field and $E|K$ an elliptic curve over the field $K$, the possible torsion groups are:
\begin{itemize}
\item[-] $\Z/n\Z$, where $1\le n\le 16$, $n=18$,
\item[-] $\Z/2\Z\times \Z/2mZ$, where $1\le m\le 6$,
\item[-] $\Z/3\Z \times \Z/3k\Z$, where $k=1,2$, $K=\Q(\sqrt{-3})$,
\item[-] $\Z/4\Z \times \Z/4\Z$, for $K=\Q(i)$.
\end{itemize}
First we construct curves of relatively large rank with torsion groups $\Z/4\Z\times \Z/4\Z$, $\Z/3\Z\times \Z/6\Z$ and $\Z/3\Z\times \Z/3\Z$.
For the first
two groups, curves of rank at least 3 were already known, and for the third group,
the already known result was that there exists a curve with rank $\ge2$.
In the thesis we construct an elliptic curve over $\Q(i)$ with torsion group $\Z/4\Z\times \Z/4\Z$ and rank equal to 7
and a family of elliptic curves with the same torsion and rank $\ge 2$.
In the case of elliptic curves over the quadratic field $\Q(\sqrt{-3})$, we construct an elliptic curve with torsion group
$\Z/3\Z\times \Z/3\Z$ and rank equal to 7 and an elliptic curve with torsion group $\Z/3\Z\times \Z/6\Z$ and rank equal to 6.
Mestre's conditional upper bound for rank plays a significant role in the methods we use, so it is studied in more detail.
For torsion groups
$\Z/2\Z\times \Z/10\Z$, $\Z/2\Z\times \Z/12\Z$, $\Z/15\Z$, $\Z/11\Z$ and $\Z/14\Z$
we study elliptic curves of positive rank over the quadratic field $\Q(\sqrt{d})$,
where the absolute value of the discriminant of the quadratic field is minimal.
Specifically, we construct curves of positive rank with torsion groups $\Z/2\Z\times \Z/10\Z$, $\Z/2\Z\times \Z/12\Z$ and $\Z/15\Z$
over the fields $\Q(\sqrt{-2})$, $\Q(\sqrt{13})$ and $\Q(\sqrt{-7})$, respectively, and we determine curves of conditionally positive rank over
the fields $\Q(\sqrt{-7})$ and $\Q(\sqrt{3})$ for torsion groups $\Z/11\Z$ and $\Z/14\Z$, respectively.
We also determine new families of elliptic curves with torsion groups $\Z/2\Z\times \Z/10\Z$ and $\Z/2\Z\times \Z/12\Z$.
Additionally, we construct elliptic curves with a fixed torsion group and maximal rank over quadratic fields
$\Q(\sqrt{d})$, $|d|\le 10^{100}$, where $|d|$ is minimal, and we give an overview of current results.
Ključne riječi
Hrčak ID:
74906
URI
Datum izdavanja:
21.12.2011.
Posjeta: 819 *