\contentsline {section}{\tocsection {}{}{Introduction}}{1}{section*.2} \contentsline {section}{\tocsection {}{1}{Cycles and the Griffiths group}}{5}{section.1} \contentsline {section}{\tocsection {}{2}{Coniveau and the refined Bloch-Beilinson conjecture}}{7}{section.2} \contentsline {subsection}{\tocsubsection {}{2.1}{The refined conjecture}}{7}{subsection.2.1} \contentsline {subsection}{\tocsubsection {}{2.2}{Abel-Jacobi}}{8}{subsection.2.2} \contentsline {subsection}{\tocsubsection {}{2.3}{The Bloch-Kato exponential and logarithm}}{8}{subsection.2.3} \contentsline {subsection}{\tocsubsection {}{2.4}{Coniveau and the Hodge filtration}}{9}{subsection.2.4} \contentsline {section}{\tocsection {}{3}{\$p\$-adic \$L\$-functions}}{12}{section.3} \contentsline {subsection}{\tocsubsection {}{3.1}{The Katz \$p\$-adic \$L\$-function}}{12}{subsection.3.1} \contentsline {subsection}{\tocsubsection {}{3.2}{\$p\$-adic Rankin \$L\$-series}}{12}{subsection.3.2} \contentsline {subsection}{\tocsubsection {}{3.3}{Factorization of the \$p\$-adic Rankin \$L\$-series}}{13}{subsection.3.3} \contentsline {section}{\tocsection {}{4}{Generalized Heegner cycles and coniveau}}{14}{section.4} \contentsline {subsection}{\tocsubsection {}{4.1}{Generalized Heegner cycles}}{14}{subsection.4.1} \contentsline {subsection}{\tocsubsection {}{4.2}{Fields of definition}}{17}{subsection.4.2} \contentsline {subsection}{\tocsubsection {}{4.3}{Cycles associated with Grossencharacters}}{18}{subsection.4.3} \contentsline {section}{\tocsection {}{5}{Miscellany}}{21}{section.5} \contentsline {subsection}{\tocsubsection {}{5.1}{A non-CM example}}{21}{subsection.5.1} \contentsline {subsection}{\tocsubsection {}{5.2}{The Fermat quartic revisited}}{24}{subsection.5.2} \contentsline {section}{\tocsection {Appendix}{A}{Tables}}{28}{section.A} \contentsline {section}{\tocsection {Appendix}{B}{Integrality of modular forms via intersection theory \\ by Brian Conrad }}{30}{section.B} \contentsline {subsection}{\tocsubsection {}{B.1}{Introduction}}{30}{subsection.B.1} \contentsline {subsubsection}{\tocsubsubsection {}{B.1.1}{Integrality and \$q\$-expansions}}{30}{subsubsection.B.1.1} \contentsline {subsubsection}{\tocsubsubsection {}{B.1.2}{Moduli stacks}}{30}{subsubsection.B.1.2} \contentsline {subsubsection}{\tocsubsubsection {}{B.1.3}{Results}}{31}{subsubsection.B.1.3} \contentsline {subsection}{\tocsubsection {}{B.2}{Intersection-theoretic bounds}}{32}{subsection.B.2} \contentsline {subsubsection}{\tocsubsubsection {}{B.2.1}{Elimination of \$q\$-expansions}}{32}{subsubsection.B.2.1} \contentsline {subsubsection}{\tocsubsubsection {}{B.2.2}{Divisor and intersection theory formalism}}{33}{subsubsection.B.2.2} \contentsline {subsubsection}{\tocsubsubsection {}{B.2.3}{Geometric calculations}}{39}{subsubsection.B.2.3} \contentsline {subsection}{\tocsubsection {}{B.3}{Explicit formulas}}{41}{subsection.B.3} \contentsline {subsubsection}{\tocsubsubsection {}{B.3.1}{Numerology for \$X_0(N)\$}}{41}{subsubsection.B.3.1} \contentsline {subsubsection}{\tocsubsubsection {}{B.3.2}{Multiplicity bounds for \$\Gamma _0(N)\$}}{43}{subsubsection.B.3.2} \contentsline {section}{\tocsection {Appendix}{}{References}}{46}{section*.3}