Computations of cyclotomic lattices.
Computations of Galois representations associated to modular forms of level one
We propose an improved algorithm for computing mod ℓ Galois representations associated to a cusp form f of level one. The proposed method allows us to explicitly compute the case with ℓ = 29 and f of weight k = 16, and the cases with ℓ = 31 and f of weight k = 12,20,22. All the results are rigorously proved to be correct. As an example, we will compute the values modulo 31 of Ramanujan's tau function at some huge primes up to a sign. Also we will give an improved uper bound on...
Computations of Iwasawa invariants and
Computations of Witt Groups of Finite Groups.
Computations with Witt vectors of length
In this paper we describe how to perform computations with Witt vectors of length in an efficient way and give a formula that allows us to compute the third coordinate of the Greenberg transform of a polynomial directly. We apply these results to obtain information on the third coordinate of the -invariant of the canonical lifting as a function on the -invariant of the ordinary elliptic curve in characteristic .
Computer-aided serendipity
Computer-free evaluation of an infinite double sum via Euler sums.
Computing all elements of given index in sextic fields with a cubic subfield
Computing all monogeneous mixed dihedral quartic extensions of a quadratic field
Let be a given real quadratic field. We give a fast algorithm for determining all dihedral quartic fields with mixed signature having power integral bases and containing as a subfield. We also determine all generators of power integral bases in . Our algorithm combines a recent result of Kable [9] with the algorithm of Gaál, Pethö and Pohst [6], [7]. To illustrate the method we performed computations for
Computing fundamental domains for Fuchsian groups
We exhibit an algorithm to compute a Dirichlet domain for a Fuchsian group with cofinite area. As a consequence, we compute the invariants of , including an explicit finite presentation for .
Computing Galois groups by means of Newton polygons
Computing Hecke eigenvalues below the cohomological dimension.
Computing higher rank primitive root densities
We extend the "character sum method" for the computation of densities in Artin primitive root problems given by Lenstra and the authors to the situation of radical extensions of arbitrary rank. Our algebraic set-up identifies the key parameters of the situation at hand, and obviates the lengthy analytic multiplicative number theory arguments that used to go into the computation of actual densities. It yields a conceptual interpretation of the formulas obtained, and enables us to extend their range...
Computing Igusa's local zeta functions of univariate polynomials, and linear feedback shift registers.
Computing integral points on elliptic curves
Computing integral points on Mordell's elliptic curves.
Computing Iwasawa modules of real quadratic number fields
Computing modular degrees using -functions
We give an algorithm to compute the modular degree of an elliptic curve defined over . Our method is based on the computation of the special value at of the symmetric square of the -function attached to the elliptic curve. This method is quite efficient and easy to implement.
Computing periods of cusp forms and modular elliptic curves.
Computing powers of two generalizations of the logarithm.