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 ⇤ ← Revision 10 as of 2007-01-22 13:22:31 → Size: 3949 Editor: IftikharBurhanuddin Comment: ← Revision 11 as of 2007-01-22 17:30:48 → ⇥ Size: 3100 Editor: wstein Comment: Deletions are marked like this. Additions are marked like this. Line 10: Line 10: Line 11: Line 12: {{{ {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}{ Addition} {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}}}} *Arithmetic in Global Commutative Rings     *The ring ${Z}$ of Integers     *The ring ${Q}$ of Rational Numbers     *Arbitrary Precision Real (and Complex) Numbers     *Univariate Polynomial Rings     *Number Fields     *Multivariate Polynomial Rings  *Arithmetic in Local Commutative Rings     *Univariate Power series rings     *$p$-adic numbers  *Linear Algebra     *Arithmetic of Vectors          *Addition          *Scalar Multiplication          *Vector times Matrix     *Rational reconstruction of a matrix     *Echelon form          *Echelon form over Finite Field          *Echelon form over ${Q}$          *Echelon form over Cyclotomic Fields          *Echelon form (Hermite form) over ${Z}$     *Kernel          *Kernel over Finite Field          *Kernel over ${Q}$          *Kernel over ${Z}$     *Matrix multiplication          *Matrix multiplication over Finite Fields          *Matrix multiplication over ${Z}$          *Matrix multiplication over Extensions of ${Z}$  *Noncommutative Rings  *Group Theory  *Groebner Basis Computation  *Elliptic Curves     *Generic elliptic curve operations          *Group Law          *Invariants          *Division Polynomials     *Elliptic curves over finite fields          *Order of the group $E({{F}}_{p})$          *Order of the group $E({{F}}_{q})$          *Order of a point     *Elliptic curves over ${{Q}}$ - part I          *Birch and Swinnerton-Dyer Conjecture          *Fourier coefficients          *Canonical height of a point          *Order of a point          *Periods          *Tate's algorithm          *Conductor and Globally minimal model          *CPS height bound          *Torsion subgroup          *Nagell-Lutz          *An $l$-adic algorithm          *Another $l$-adic algorithm          *Mordell-Weil via 2-descent          *Saturation          *Heegner points          *Heegner discriminants          *Heegner Hypothesis          *Heegner point index and height     *Elliptic curves over ${{Q}}$ - part II          *Root number          *Special values of L-series          *Sha bound          *Isogenies          *Attributes of primes          *$p$-adic height          *Modular Degree          *Modular Parameterization  *Hyperelliptic Curves  *Modular Forms     *Presentation of spaces of modular symbols     *Hecke operators on modular symbols     *Decomposition of spaces under the Hecke operators     *Trace formulas  *Computation of tables     *Elliptic curves     *Modular forms     *Number fields  *Cryptography  *Coding Theory  *Constants, functions and numerical computation

# 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

• Arithmetic in Global Commutative Rings
• The ring {Z} of Integers

• The ring {Q} of Rational Numbers

• Arbitrary Precision Real (and Complex) Numbers
• Univariate Polynomial Rings
• Number Fields
• Multivariate Polynomial Rings
• Arithmetic in Local Commutative Rings
• Univariate Power series rings

• Linear Algebra
• Arithmetic of Vectors
• Scalar Multiplication
• Vector times Matrix
• Rational reconstruction of a matrix
• Echelon form
• Echelon form over Finite Field
• Echelon form over {Q}

• Echelon form over Cyclotomic Fields
• Echelon form (Hermite form) over {Z}

• Kernel
• Kernel over Finite Field
• Kernel over {Q}

• Kernel over {Z}

• Matrix multiplication
• Matrix multiplication over Finite Fields
• Matrix multiplication over {Z}

• Matrix multiplication over Extensions of {Z}

• Noncommutative Rings
• Group Theory
• Groebner Basis Computation
• Elliptic Curves
• Generic elliptic curve operations
• Group Law
• Invariants
• Division Polynomials
• Elliptic curves over finite fields
• Order of the group E({{F}}_{p})

• Order of the group E({{F}}_{q})

• Order of a point
• Elliptic curves over {{Q}} - part I

• Birch and Swinnerton-Dyer Conjecture
• Fourier coefficients
• Canonical height of a point
• Order of a point
• Periods
• Tate's algorithm
• Conductor and Globally minimal model
• CPS height bound
• Torsion subgroup
• Nagell-Lutz

• Mordell-Weil via 2-descent
• Saturation
• Heegner points
• Heegner discriminants
• Heegner Hypothesis
• Heegner point index and height
• Elliptic curves over {{Q}} - part II

• Root number
• Special values of L-series
• Sha bound
• Isogenies
• Attributes of primes

• Modular Degree
• Modular Parameterization
• Hyperelliptic Curves
• Modular Forms
• Presentation of spaces of modular symbols
• Hecke operators on modular symbols
• Decomposition of spaces under the Hecke operators
• Trace formulas
• Computation of tables
• Elliptic curves
• Modular forms
• Number fields
• Cryptography
• Coding Theory
• Constants, functions and numerical computation

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