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0.0, 0.99328859060402563) \newrgbcolor{mycolor}{0.01 0.00 0.99}\mycolor %Solid dot at (-0.134,0.117) of radius 0.025 \pscircle*[linecolor=mycolor](-0.134,0.117){\dotsize} \endpspicture %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% } \page{ \heading{Mordell's Theorem} \vspace{2ex} \psset{unit=1.0} \pspicture(0,0)(0,0) \eps{17.5}{0.9}{0.25}{pics/mordell1} \endpspicture {\dred Theorem (Mordell).} The group $E(\Q)$ of rational points on an elliptic curve is a {\dgreen finitely generated abelian group}, so $$ E(\Q) \cong \Z^r \oplus T, $$ with $T=E(\Q)_{\rm tor}$ finite. \vspace{2ex} Mazur classified the possibilities for $T$. \vfill \rd{Folklore conjecture:} $r$ can be arbitrary. (The record is $\geq 24$.) } % end page ------ \page{ \heading{Motivating Problem} \vfill \bd{Motivating Problem.} Verify the full BSD conjecture for specific curves. \par\noindent{}(NOTE: I am also very interested in the same question but for modular abelian varieties. I don't know a single nontrivial example of $\dim \geq 3$ that has been done to completion.) \vfill } ------ \page{ \heading{Our Strategy} \begin{enumerate} \item{}[\rd{Kolyvagin}] Study computation of Kolyvagin's bound (computation of Heegner points) \item{}[\rd{Refine}] \label{step:refine} Prove a refinement of \underline{Kolyvagin's bound} on $\#\Sha(E)$ that is suitable for computation. \item{}[\rd{Kato}] Study computation of Kato's bound (Grigorov) when $r_\an=0$. \vspace{-2ex} \end{enumerate} } \page{ \heading{What Happens} I will assume you know about Heegner points now. Go through bsdalg paper and summarize ``what happens''. computing images of Galois reps. computing bounds on heights of Heegner points using search and CPS refining Kolyvagin in some important cases $3$-descent every case gets dealt with one way or another } \page{ \heading{An Example} \vfill The elliptic curve $E$ called 141A and given by $y^2 + y = x^3 + x^2 - 12x + 2$ has rank 1 and $c_3 = 7$. Our computation using Kolyvagin implies that $$ \#\Sha(E) \mid 49. $$ The representation $\rho_{E,7}$ is surjective, but~$E$ has rank~$1$.\\ \vfill %\begin{itemize} %\item{}[Visibility?] %The Jacobian $J_0(47)$ is of rank %$0$ and is simple of dimension $4$, and we find that $E[7]$ sits in %the old subvariety of $J_0(3\cdot 47)$. %Hope: Proving %something about the Shafarevich-Tate group of the simple rank $0$ abelian %variety $J_0(47)$ will imply something about $\Sha(E)[7]$. %Note that $L(J_0(47),1)/\Omega = 16/23$. %\vfill %\item{}[$p$-Adic Approach?] %\rd{Greenberg's suggestion:} %Compute a $p$-adic $L$-function, a $p$-adic regulator, and use %theorems of Kato and Peter Schneider to show that $7\nmid %\#\Sha(E)$. I hope to do this soon. %\end{itemize} } \end{document} \page{ \heading{More Examples} \begin{itemize} \item 190A1: We have $190=2\cdot 5\cdot 19$ and $c_{2}=11$. There is a $4$-dimensional abelian variety over rank $0$ and level $95$ with $\Sha[11]$ trivial that contains $E[11]$. \item 214A1: We have $214=2\cdot 107$ and $c_{2}=7$. There is a rank $0$ simple abelian variety over level $107$ and dimension $7$ that contains $E[7]$. \item 674C1: We have $214=2\cdot 337$ and $c_{2}=31$. For this one, there is a rank $0$ simple abelian variety of level $337$ and dimension $15$ that contains $E[31]$ and according to BSD has trivial $\Sha[31]$. \end{itemize} }