Chebyshev's bias.
CM liftings of supersingular elliptic curves
Assuming GRH, we present an algorithm which inputs a prime and outputs the set of fundamental discriminants such that the reduction map modulo a prime above from elliptic curves with CM by to supersingular elliptic curves in characteristic is surjective. In the algorithm we first determine an explicit constant so that implies that the map is necessarily surjective and then we compute explicitly the cases .
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...
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 the modular degree of an elliptic curve.
Computing the summation of the Möbius function.
Convergence acceleration of alternating series.