\relax 
\citation{darmon-bsd}
\citation{cime-1997}
\citation{rubin:main-conjectures}
\citation{gross-zagier}
\citation{kolyvagin:mordellweil}
\citation{cremona-mazur}
\citation{breuil-conrad-diamond-taylor}
\citation{diamond-im}
\citation{mazur:rational}
\citation{kolyvagin-logachev:totallyreal}
\@writefile{toc}{\contentsline {chapter}{\numberline {1}The Birch and Swinnerton-Dyer conjecture}{2}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\newlabel{chap:bsd}{{1}{2}}
\@writefile{toc}{\contentsline {section}{\numberline {1.1}The BSD conjecture}{2}}
\citation{milne:abvars}
\citation{mazur:rational}
\citation{cremona:algs}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.1.1}The ratio $L(A,1)/\Omega _A$}{3}}
\newlabel{thm:ratpart}{{1.2}{3}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.1.2}Torsion subgroup}{3}}
\citation{mazur:eisenstein}
\citation{mestre-oesterle:crelle}
\citation{ribet:modreps}
\citation{merel:weil}
\citation{bloch-kato}
\citation{dummigan:cp}
\citation{scholl:motivesinvent}
\citation{kolyvagin:structureofsha}
\citation{scholl:kato}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.1.3}Tamagawa numbers}{4}}
\newlabel{thm:tamagawa}{{1.5}{4}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.1.4}Upper bounds on $\#\unhbox \voidb@x \hbox {\cyr X}(A)$}{4}}
\citation{kolyvagin:mordellweil}
\citation{kolyvagin-logachev:totallyreal}
\citation{rubin:book}
\citation{cremona-mazur}
\citation{mazur:visthree}
\@writefile{toc}{\contentsline {subsubsection}{Kolyvagin's bounds}{5}}
\@writefile{toc}{\contentsline {subsubsection}{Kato's bounds}{5}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.1.5}Lower bounds on $\#\unhbox \voidb@x \hbox {\cyr X}(A)$}{5}}
\citation{ribet:raising}
\citation{cremona-mazur}
\citation{cremona-mazur}
\@writefile{toc}{\contentsline {subsubsection}{Invisible elements of $\#\unhbox \voidb@x \hbox {\cyr X}(A^{\vee })$}{6}}
\@writefile{toc}{\contentsline {subsubsection}{Visibility at higher level}{6}}
\@writefile{toc}{\contentsline {subsubsection}{Visibility in some Jacobian}{6}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.1.6}Motivation for considering abelian varieties}{6}}
\@writefile{toc}{\contentsline {section}{\numberline {1.2}Existence of nontrivial visible elements of $\unhbox \voidb@x \hbox {\cyr X}(A)$}{6}}
\citation{shafarevich:exp}
\newlabel{thm:shaexists}{{1.8}{7}}
\citation{milne:duality}
\citation{neronmodels}
\citation{neronmodels}
\citation{milne:etale}
\citation{milne:duality}
\citation{neronmodels}
\citation{neronmodels}
\citation{milne:duality}
\citation{stein:hecke}
\citation{magma}
\@writefile{toc}{\contentsline {section}{\numberline {1.3}Description of tables}{9}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.3.1}Notation}{9}}
\newlabel{sec:optquo-notation}{{1.3.1}{9}}
\citation{brumer:rank}
\citation{brumer:rank}
\citation{brumer:rank}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.3.2}Table\nobreakspace  {}1.2\hbox {}: Shafarevich-Tate groups at prime level}{10}}
\@writefile{toc}{\contentsline {subsubsection}{Notation}{10}}
\@writefile{toc}{\contentsline {subsubsection}{Ranks of the explanatory factors}{10}}
\@writefile{toc}{\contentsline {subsubsection}{Discussion of the data}{10}}
\citation{agashe}
\citation{cremona-mazur}
\citation{brumer:rank}
\citation{cremona-mazur}
\citation{cremona-mazur}
\citation{cremona-mazur}
\@writefile{toc}{\contentsline {subsubsection}{Errata to Brumer's paper}{11}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.3.3}Tables\nobreakspace  {}1.3\hbox {}--1.6\hbox {}: New visible Shafarevich-Tate groups}{11}}
\@writefile{toc}{\contentsline {subsubsection}{Notation}{11}}
\@writefile{toc}{\contentsline {subsubsection}{Remarks on the data}{11}}
\citation{cremona-mazur}
\citation{cremona-mazur}
\citation{cremona-mazur}
\@writefile{lot}{\contentsline {table}{\numberline {1.1}{\ignorespaces Odd invisible $|\unhbox \voidb@x \hbox {\cyr X}_E|>1$, all $N\leq 5500$ (from Table\nobreakspace  {}1 of\nobreakspace  {}\cite  {cremona-mazur})}}{12}}
\newlabel{table:cm}{{1.1}{12}}
\@writefile{toc}{\contentsline {section}{\numberline {1.4}Further visibility computations}{12}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.4.1}Does $\unhbox \voidb@x \hbox {\cyr X}$ become visible at higher level?}{12}}
\newlabel{sec:higherlevel}{{1.4.1}{12}}
\@writefile{toc}{\contentsline {subsubsection}{How we found the explanatory curves}{12}}
\citation{cremona-mazur}
\citation{ribet:raising}
\@writefile{toc}{\contentsline {subsubsection}{2849A}{13}}
\@writefile{toc}{\contentsline {subsubsection}{4343B}{13}}
\@writefile{toc}{\contentsline {subsubsection}{5389A}{13}}
\citation{cremona:algs}
\citation{mestre:graphs}
\citation{zagier:parametrizations}
\citation{mazur:eisenstein}
\citation{cremona:algs}
\citation{cremona:algs}
\citation{cremona:algs}
\citation{cremona:algs}
\@writefile{toc}{\contentsline {subsubsection}{3364C, 4229A, 5073D}{14}}
\@writefile{toc}{\contentsline {subsubsection}{4194N, 5054C}{14}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.4.2}Positive rank example}{14}}
\@writefile{lot}{\contentsline {table}{\numberline {1.2}{\ignorespaces Shafarevich-Tate groups at prime level. (The entries in the columns ``mod deg'' and ``$\#\unhbox \voidb@x \hbox {\cyr X}_{\unhbox \voidb@x \hbox {\fontsize  6{7pt}\selectfont  \rm  an}}$'' are only really the odd parts of ``mod deg'' and ``$\#\unhbox \voidb@x \hbox {\cyr X}_{\unhbox \voidb@x \hbox {\fontsize  6{7pt}\selectfont  \rm  an}}$''.)}}{15}}
\newlabel{table:primesha}{{1.2}{15}}
\@writefile{lot}{\contentsline {table}{\numberline {1.3}{\ignorespaces New visible Shafarevich-Tate groups}}{16}}
\newlabel{table:newvis}{{1.3}{16}}
\@writefile{lot}{\contentsline {table}{\numberline {1.4}{\ignorespaces Explanatory factors}}{17}}
\newlabel{table:explain}{{1.4}{17}}
\@writefile{lot}{\contentsline {table}{\numberline {1.5}{\ignorespaces Factorizations}}{17}}
\newlabel{table:factor}{{1.5}{17}}
\@writefile{lot}{\contentsline {table}{\numberline {1.6}{\ignorespaces Component groups}}{18}}
\newlabel{table:shacompgps}{{1.6}{18}}
\@setckpt{bsd}{
\setcounter{page}{19}
\setcounter{equation}{0}
\setcounter{enumi}{0}
\setcounter{enumii}{0}
\setcounter{enumiii}{0}
\setcounter{enumiv}{0}
\setcounter{footnote}{1}
\setcounter{mpfootnote}{0}
\setcounter{part}{0}
\setcounter{chapter}{1}
\setcounter{section}{4}
\setcounter{subsection}{2}
\setcounter{subsubsection}{0}
\setcounter{paragraph}{0}
\setcounter{subparagraph}{0}
\setcounter{figure}{0}
\setcounter{table}{6}
\setcounter{parentequation}{0}
\setcounter{theorem}{8}
\setcounter{listnum}{0}
}