Differences between revisions 12 and 13
 ⇤ ← Revision 12 as of 2007-01-22 17:32:05 → Size: 3247 Editor: wstein Comment: ← Revision 13 as of 2007-01-22 17:32:47 → ⇥ Size: 3288 Editor: wstein Comment: Deletions are marked like this. Additions are marked like this. Line 9: Line 9: == SAGE: General Problems == == Parallel Implementations == Line 11: Line 11: For each of the following, make remarks about how '''specific practical implementable''' parallel algorithms could be used to enhance SAGE. For each of the following, make remarks about how '''specific practical implementable''' parallel algorithms could be used to enhance mathematics software libraries (e.g., SAGE).

# 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]

## Parallel Implementations

For each of the following, make remarks about how specific practical implementable parallel algorithms could be used to enhance mathematics software libraries (e.g., SAGE).

• 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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