Questions tagged [hyperelliptic-curves]
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Calculate reduction of Jacobian of hyperelliptic curve
Suppose I have a hyperelliptic curve of genus $2$ over $\mathbb Q$. I want to get information about its Jacobian reduction at prime $p$ (especially, in case $p=2$). Also I'm interesting in the group ...
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rational points of a hyperelliptic curve
I have the following hyperelliptic curve of genus $2$:
$$
y^2 = 561 x^6 - 41904 x^5 + 627264 x^4 + 11860992 x^3 - 197074944 x^2 + 124416^2
$$
I need to find all the rational points on this curve. ...
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Rational perfect power values of $y(y+1)$
This is hard, so I am looking for partial results and how hard it is.
Let $n>4$. Is it true that the hyperelliptic curve $x^n=y(y+1)$
doesn't have rational point with $x \ne 0$?
If necessarily ...
6
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Understanding of Tamagawa numbers of hyperelliptic curve
One's can find following definition of tamagawa numbers in Dino Lorenzini paper "Torsion and Tamagawa numbers":
Let $K$ be any discrete valuation field with ring of integers $O_K$ ,
uniformizer $...
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1
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Abelian varieties with CM
In this site, I looked at a paper of Kazuma Morita claiming the BSD conjecture for the CM case
posted on his homepage (he made a mistake three years ago for full BSD).
But, I am interested in this ...
2
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1
answer
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Canonical lifts from $\mathbb F_q$ and CM-theory
One knows that (ordinary) Jacobians of hyperelliptic curves over a finite field $\mathbb F_q$ (mostly of genus 1 (elliptic curves) and 2) are extensively studied by cryptographers, as a platform for ...
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Hyperelliptic curves imply FLT-like results
Probably this is known, but doesn't show in searches.
If a certain hyperelliptic curve has only trivial rational points,
FLT-like curve also has only trivial rationals points for fixed $n$.
Working ...
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Elementary constraints for the solutions of $z^{n-2}y(y+z)=x^n$?
Related to FLT and this question.
For natural $n > 4 $ define the curve $C_n : z^{n-2}y(y+z)=x^n$.
$C_n$ has the trivial points with $x=0$ for all $n$.
The answer in the linked question shows ...