Differences between revisions 10 and 11

MSRI 2007 Parallel Computation Problem List

SAGE-related Problems

[:msri07/threadsafety: Thread Safety of the SAGE Libraries]
[:msri07/pthread_sagex: Add Pthread support to SageX]
[: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
- 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(Fp)
  - Order of the group E(Fq)
  - 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

-  ⇤ ← Revision 10 as of 2007-01-22 13:22:31 → 
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+   ← Revision 11 as of 2007-01-22 17:30:48 → ⇥
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  Editor: wstein
  Comment:
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-{{{

 {chapter}{ Arithmetic in Global Commutative Rings}
 {section}{ The ring  $\@ m a t h b b Z$  of Integers}
 {section}{ The ring  $\@ m a t h b b Q$  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  $\@ m a t h b b Q$ }
 {subsection}{ Echelon form over Cyclotomic Fields}
 {subsection}{ Echelon form (Hermite form) over  $\@ m a t h b b Z$ }
 {section}{Kernel}
 {subsection}{ Kernel over Finite Field}
 {subsection}{ Kernel over  $\@ m a t h b b Q$ }
 {subsection}{ Kernel over  $\@ m a t h b b Z$ }
 {section}{ Matrix multiplication}
 {subsection}{ Matrix multiplication over Finite Fields}
 {subsection}{ Matrix multiplication over  $\@ m a t h b b Z$ }
 {subsection}{ Matrix multiplication over Extensions of  $\@ m a t h b b Z$ }
 {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  $E ({\@ m a t h b b F}_{p})$ }
 {subsection}{ Order of the group  $E ({\@ m a t h b b F}_{q})$ }
 {subsection}{ Order of a point}
 {section}{ Elliptic curves over  $\@ m a t h b b Q$  - 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  $\@ m a t h b b Q$  - part II}
 {subsection}{ Root number}
 {subsection}{ Special values of L-series}
 {subsection}{  $# \unhbox \voidb @ x \fontencoding OT2 \fontfamily wncyr \fontseries m \fontshape n \selectfont Sh (E)$  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

Diff for "msri07/problems"

MSRI 2007 Parallel Computation Problem List

SAGE-related Problems

Computer Algebra Problems