All Questions
Tagged with arithmetic-geometry computational-number-theory
9 questions
2
votes
0
answers
54
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Conductor of hyperelliptic curve after adding a rational root
Let $r\geq 5$ be a prime. Suppose I have a specified hyperelliptic curve $C: y^2=f(x)$ defined over $\mathbb{Q}$, where $f\in\mathbb{Q}[x]$ has degree $r$.
Note: the roots of $f$ are not rational but ...
- 83
-1
votes
1
answer
342
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Finite or polynomial number integral points clarification on Coppersmith's theorems (possibility of ellipse counter example?)
Coppersmith states if $f(x,y)$ is an irreducible bivariate with total degree $\delta$ then if he can list all roots $(X,Y)$ of the polynomial in $\mathsf{poly}(\log D,\delta)$ time if the roots ...
- 13.9k
3
votes
0
answers
171
views
Scalar multiplication via the Kummer surface of a genus $2$ curve by $\sqrt{5}$
I hope this is a good question.
Recently I worked with genus two curves $H$ that have multiplication by $[\zeta_5]\in \text{Aut}(H)$, that is, multiplication by $e^{2\pi i/5}$. This automorphism is ...
2
votes
2
answers
257
views
Reference request for function by which to compute coefficients of continued fraction of algebaic number
The simple continued fraction is in the form
$$[1;1,2,3,4,5,\dots]=1+\cfrac{1}{1+\cfrac{1}{2+\cdots}}, $$ for instance. Obviously,the coefficients $x_i$can be computed by computable function $x_i=f(i),...
- 3,094
2
votes
3
answers
476
views
How to define the input of computable function or Turing machine over real numbers
Computation or computability over $\mathbb{N}$ can be extended to computation or computability over $\mathbb{R}$ or even computation or computability over $\mathbb{C}$.The following is a formal ...
- 3,094
1
vote
0
answers
204
views
Which rational subfields are corresponding to the two dimensional subspaces of holomorphic differentials
I implemented the algorithm that Felipe Voloch's suggested in his reply to the question:
Subfields of a function field
the algorithm is here:
Subfields of a function field
I considered the ...
- 601
1
vote
0
answers
108
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Why do subspaces of the space of Global holomorphic differentials of a function field correspond to its subfields
I'm asking this question as a follow up to the Felipe Voloch's answer to this question:
Subfields of a function field
which you can read it here:
Subfields of a function field
(I just didn't have ...
- 601
0
votes
0
answers
263
views
Computing the function field of a curve given as a subvariety of the Jacobian of its cover or merely the degree of the covering
I read following paragraph from:
G. Tamme, Teilkörper höheren Geschlechts eines algebraischen Funktionenkörpers, Arch. Math. 23 (1972), 257--259
Here $C$ is a curve of genus $\ge 2$ and $J$ is the ...
- 601
11
votes
0
answers
854
views
Points of bounded height in a number field
Let $K$ be a number field of absolute degree $d$, let $B$ be a positive real number, and write $S(K, B) = \{x \in K : H(x) \leq B\}$. Here $H$ is the absolute multiplicative height of an algebraic ...
- 1,199