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Results tagged with computational-number-theory
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user 2024
Computational Number Theory is for explicit calculations or algorithms involving anything of interest to number theorists.
2
votes
Accepted
on the computation of decomposition groups
Henri Cohen's A Course in Computational Algebraic Number Theory contains quite a bit of information. Chapters 4.8, 6.2 and 6.3 combined result in algorithms that compute decomposition groups. Note tha …
- 4,593
0
votes
Upper bound on greatest prime of bad reduction for a plane curve
The primes that are "bad" in your sense will divide the number $Res_x(Res_y(f,\frac{\partial f}{\partial x}), Res_y(f,\frac{\partial f}{\partial y}))$. (If I interpreted damiano's comment correctly).
…
- 4,593
3
votes
2
answers
377
views
Upper bound on greatest prime of bad reduction for a plane curve
Background
We are given a curve with integer coefficients. I want to make a suggestion in another question (Computationally bounding a curve's genus from below?) into a deterministic algorithm for fi …
- 4,593
11
votes
1
answer
562
views
CM field to Torus to Abelian Variety?
Given a CM field we can use its maximal order (and a choice of CM type) to construct an abelian variety $\mathbb{C}^g/\Lambda$ with complex multiplication by the maximal order.
How do I (or where can …
- 4,593
7
votes
2
answers
727
views
Explicit map for Scholz reflection principle
The question is about the specific case of reflection theorems (copied straight from Franz Lemmermeyer's "Class Groups of Dihedral Extensions"):
Let $k^+ = \mathbb{Q}(\sqrt{m})$ with $m\in \mathbb …
- 4,593
10
votes
3
answers
633
views
Computationally bounding a curve's genus from below?
Background
In the course of answering another question (Infinite collection of elements of a number field with very similar annihilating polynomials) I found myself with a curve, that if it had a pos …
- 4,593
5
votes
What does the computer suggest about the parity of p(n), for n in a fixed arithmetic progres...
The answer to your very last question is yes. As for the rest:
Computing the parity up to N is, in theory, quasi linear by using Pentagonal Number Theorem: compute the inverse of the pentagonal numbe …
- 4,593
2
votes
Accepted
Speeding the quadratic sieve with an oracle
If we know the factorization of $N$ then we can take square roots of small numbers that are quadratic residues mod all primes dividing $N$. Knowing a partial factorization of size $M\sim N^\alpha$, sq …
- 4,593