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