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# MSRI 2007 Parallel Computation Problem List

## 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

## John McKay CHALLENGE system of polynomial equations

msri07/problems (last edited 2008-11-14 13:42:04 by localhost)