# Questions tagged [hyperelliptic-curves]

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### Computing the class group of a quadratic function field

I am asking for a reference in which I can find tools to answer questions like the following: Let $K=\mathbb{F}_q(X)$ be a rational function field over the finite field with $q$ elements. Let $E/K$ be ...
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### Is the Jacobian isogenous over $\mathbb{F}_p$ to the direct product of the elliptic curves?

Let $\mathbb{F}_p$ be a finite field such that $p \equiv 1 \ (\mathrm{mod} \ 3)$ and $p \equiv 3 \ (\mathrm{mod} \ 4)$. Consider the Jacobian of the hyperelliptic curve $C\!: y^2 = (x^3 + b)(x^3-b)$, ...
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### Visualizing a normalized hyper-elliptic curve as a honey-flow on a multi-doughnut

Trying to visualize in 3D the normalized model $y^2=f(x)$ of a hyper-elliptic curve AND the degree two ramified cover $(X,Y)\mapsto X$, $\mathbb{C}\times \mathbb{C}\to \mathbb{C}$. Is this projection ...
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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 ...
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### 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 ...
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### How can I describe the monodromy of this variation of singular curves?

Consider the family of singular hyperelliptic curves $$y^2 - x(x-1)^2(x-2)(x-3)(x-4)(x-t)$$ over $\mathbb{A}^1_t$. Over a generic point the fiber is a genus three curve where one of the genera comes ...
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### Is there a finite number of supersingular genus 2 curves?

Consider an algebraically closed field $k$ of prime characteristic. It is widely known that up to $k$-isomorphism there is a finite number of supersingular elliptic curves over $k$. Let $C$ be a ...
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### Prime divisors on the Jacobian of a genus 2 curve over $\mathbb{F}_q$ under the $n$ map

Let $H$ be a hyperelliptic curve geometrically irreducible of genus 2 over $\mathbb{F}_q$ with a rational point $\infty$ given by the model $y^2=f(x)$, where $f$ is monic of degree 5. Consider the ...
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Does there exist a hyperelliptic curve $X$ of genus $g\geq 2$ over the complex numbers such that $X$ has a hyperelliptic quotient $X\to Y$ (in the sense that $Y$ is hyperelliptic and the morphism $X\... 1answer 379 views ### Easiest example where field of definition is not field of moduli There are many examples of varieties over$\overline{\mathbb Q}$whose field of moduli is$\mathbb Q$but which can't be defined over$\mathbb Q$. What is the easiest such example? It should be a ... 4answers 504 views ### Equivalent binary forms Two binary forms$f, g \in k[x, y]$are equivalent when there exists an$M \in GL_2 (k)$such that$f^M = g$. For simplicity we take$k$such that$char (k) =0$and$k=\bar k$. The equivalence ... 0answers 163 views ### Computing Tamagawa numbers for jacobians of hyperelliptic curves Do exist some computational approach to calculation of Tamagawa number for the jacobian of hyperelliptic curve at prime$p$? As followed from this question one can compute$\Phi(\overline{\mathbb F}...
Let $C \to \mathbf P^1$ be a hyperelliptic curve of genus $g \ge 2$ obtained as a double cover of $\mathbf P^1$ branched at $r$ points. Let $\tilde U\subset C$ be its open subset obtained by removing ...
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 \$...