# MSRI 2007 Parallel Computation Problem List

• [:msri07/anlist: Implementation in SAGE parallel computation of elliptic curve a_p for all p up to some bound]

## Computer Algebra Problems

 {chapter}{ Arithmetic in Global Commutative Rings}
{section}{ The ring $\@mathbb {Z}$ of Integers}
{section}{ The ring $\@mathbb {Q}$ of Rational Numbers}
{section}{ Arbitrary Precision Real (and Complex) Numbers}
{section}{ Univariate Polynomial Rings}
{section}{ Number Fields}
{section}{ Multivariate Polynomial Rings}
{chapter}{ Arithmetic in Local Commutative Rings}
{section}{ Univariate Power series rings}
{section}{ $p$-adic numbers}
{chapter}{ Linear Algebra}
{section}{ Arithmetic of Vectors}
{subsection}{ Scalar Multiplication}
{subsection}{ Vector times Matrix}
{section}{ Rational reconstruction of a matrix}
{section}{ Echelon form}
{subsection}{ Echelon form over Finite Field}
{subsection}{ Echelon form over $\@mathbb {Q}$}
{subsection}{ Echelon form over Cyclotomic Fields}
{subsection}{ Echelon form (Hermite form) over $\@mathbb {Z}$}
{section}{Kernel}
{subsection}{ Kernel over Finite Field}
{subsection}{ Kernel over $\@mathbb {Q}$}
{subsection}{ Kernel over $\@mathbb {Z}$}
{section}{ Matrix multiplication}
{subsection}{ Matrix multiplication over Finite Fields}
{subsection}{ Matrix multiplication over $\@mathbb {Z}$}
{subsection}{ Matrix multiplication over Extensions of $\@mathbb {Z}$}
{chapter}{ Noncommutative Rings}
{chapter}{ Group Theory}

## Arithmetic Geometry Problems

 {chapter}{ Groebner Basis Computation}
{chapter}{ Elliptic Curves}
{section}{ Generic elliptic curve operations}
{subsection}{ Group Law}
{subsection}{ Invariants}
{subsection}{ Division Polynomials}
{section}{ Elliptic curves over finite fields}
{subsection}{ Order of the group $E({\@mathbb {F}}_{p})$}
{subsection}{ Order of the group $E({\@mathbb {F}}_{q})$}
{subsection}{ Order of a point}
{section}{ Elliptic curves over ${\@mathbb {Q}}$ - part I}
{subsection}{ Birch and Swinnerton-Dyer Conjecture}
{subsection}{ Fourier coefficients}
{subsection}{ Canonical height of a point}
{subsection}{ Order of a point}
{subsection}{ Periods}
{subsection}{ Tate's algorithm}
{subsection}{ Conductor and Globally minimal model}
{subsection}{ CPS height bound}
{subsection}{ Torsion subgroup}
{subsubsection}{Nagell-Lutz}
{subsubsection}{An $l$-adic algorithm}
{subsubsection}{Another $l$-adic algorithm}
{subsection}{ Mordell-Weil via 2-descent}
{subsection}{ Saturation}
{subsection}{ Heegner points}
{subsubsection}{Heegner discriminants}
{subsubsection}{Heegner Hypothesis}
{subsubsection}{Heegner point index and height}
{section}{ Elliptic curves over ${\@mathbb {Q}}$ - part II}
{subsection}{ Root number}
{subsection}{ Special values of L-series}
{subsection}{ $\# {\unhbox \[email protected] \hbox {{\fontencoding {OT2}\fontfamily {wncyr}\fontseries {m}\fontshape {n}\selectfont Sh (E)$ bound}

• {subsection}{ Isogenies} {subsection}{ Attributes of primes}

{subsection}{ p-adic height} {subsection}{ Modular Degree} {subsection}{ Modular Parameterization} {chapter}{ Hyperelliptic Curves} {chapter}{ Modular Forms} {section}{ Presentation of spaces of modular symbols} {section}{ Hecke operators on modular symbols} {section}{ Decomposition of spaces under the Hecke operators} {section}{ Trace formulas}

}}}

## Other Topics Problems

 {chapter}{ Computation of tables}
{section}{ Elliptic curves}
{section}{ Modular forms}
{section}{ Number fields}
{chapter}{ Cryptography}
{chapter}{ Coding Theory}
{chapter}{ Constants, functions and numerical computation}