All Questions
Tagged with rational-points reference-request
8 questions
4
votes
2
answers
343
views
Algorithm for computing rational points if the rank of Jacobian is 0
Is there a general algorithm that can compute in finite time all rational points on any curve of genus $g\geq 2$ whose Jacobian has rank $0$?
If not, for what special cases such algorithm is known? ...
- 7,182
7
votes
1
answer
296
views
Reference request. Finiteness of the Selmer group
Let $K$ be a global field (ie either a number field or the function field of a curve over a finite field). Let $A,B$ be abelian varieties over $K$ and let $\phi:A\to B$ be an isogeny. Associated with $...
- 3,076
-1
votes
1
answer
315
views
Bounds for the number of points on projective hyperelliptic curves over finite fields
Let $C$ be projective hyperelliptic curve over finite field $K$.
What are bounds for the number of points $\#C(K)$?
The Hasse-Weil bound requires smooth curves, and hyperelliptic curves are
not smooth ...
- 25.4k
9
votes
3
answers
737
views
Around the diophantine equation $\frac{a}{2b+3c}+\frac{b}{2c+3a}+\frac{c}{2a+3b}=\text{odd integer}$, over positive integers
I am interested to know if a similar theorem that shows this answer of the post
Estimating the size of solutions of a diophantine equation (this MathOverflow, January 5th 2016) is feasible for a ...
7
votes
1
answer
508
views
What is the exact statement about uniform boundedness of rational points on curves of genus greater than one? Singular points can be unbounded
According to several sources, it is conjectured (or at least believed)
that the rational points of curves over the rationals of genus $g > 1$
are uniformly bounded by $g$. E.g. here p. 1.
Assuming ...
- 25.4k
2
votes
0
answers
120
views
Benchmark problems for computing rational points on varieties
Are there standard benchmark problem sets used for empirically evaluating algorithms designed for computing rational points on (various classes of) algebraic varieties?
If so, could you please point ...
24
votes
3
answers
3k
views
Integer-distance sets
Let $S$ be a set of points in $\mathbb{R}^d$; I am especially interested in $d=2$.
Say that $S$ is an integer-distance set if every pair of points in $S$ is separated
by an integer Euclidean distance.
...
- 151k
6
votes
1
answer
667
views
Pick's Theorem for rational points of bounded height
I wonder if the various lattice-point theorems, such as
Pick's Theorem or
Minkowski's Lattice Theorem,
have been generalized to the collection of points
with rational coordinates no more than height ...
- 151k