Questions tagged [hyperelliptic-curves]
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Automorphism group of symmetric square
Say I have a hyperelliptic curve without any automorphism beyond the hyperelliptic involution.
Is it possible for its symmetric square to obtain new automorphisms beyond the one induced by the ...
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how to derive this elliptic integral?
I am reading the article arXiv: 2207.09961, there are some interesting elliptic integrals, i.e. the formula (3.7) and (3.8). You can also see this image
where $p_0(z)=\sqrt{-Q_0(z)}$ and $Q_0(z)=-\...
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If a genus 2 curve has no $k$-rational points, can it have a $k$-rational divisor class of degree 1?
Let $C$ be a smooth projective geometrically connected curve of genus 2 defined over a number field $k$. Here are some definitions:
The index $I$ of a curve $C$ is the greatest common divisor of all ...
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Can I calculate congruent zeta function of given hyperelliptic curve by hand?
How can I calculate the numerator of congruent zeta function of given hyperelliptic curve ?
For example, let $C:y^2=(x^2+1)(x^4-8x^3+2x^2+8x+1)$.
numerator of congruent zeta function mod$23$ of this ...
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Missing generator for $H^0(C, \, \omega_C^{\otimes 2})$, with $C$ is hyperelliptic of genus $3$
This is probably very classical and well-known, but I could not find the answer in the literature, so let me ask it here.
Let $C$ be a hyperelliptic curve of genus $3$, defined over the complex ...
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Finding the $K=\mathbb{Q}(\sqrt{6})$-rational points on the twist of $X_{0}(26)$
Let $K=\mathbb{Q}(\sqrt{6})$. I am looking to determine all $K$-rational points on the curve
$$C: y^{2}=3x^6-24x^5+24x^4-54x^3+24x^2-24x+3.$$
More precisely, $C$ is a twist of the modular curve $X_{0}(...
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Morphisms from plane curves to hyperelliptic curves
Consider a plane curve $\mathcal{C}$ of degree $d$. We know that if a morphism $\varphi$ from $\mathcal{C}$ to a curve of genus $g\geq 2$ exists, then $\deg \varphi \leq (g'-1)/(g-1)$ where $g'$ is ...
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From a physicist: How do I show certain superelliptic curves are also hyperelliptic?
As the title suggests, I am a physicist and have a question about how to show certain superelliptic curves are also hyperelliptic. The superelliptic Riemann surfaces in question has the form $$w^n = \...
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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 ...
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Richelot isogenies in characteristic $2$
I am interested in Richelot isogenies between ordinary abelian surfaces in characteristic $2$. If I am not mistaken, the corresponding theory is developed in Article "J.-B. Bost, J.-F. Mestre, ...
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Are there non-trivial $\mathbb{F}_q$-covers of the j-invariant 0 elliptic curve by a hyperelliptic or cyclic trigonal curve?
Consider the ordinary elliptic curve $E\!: y^2 = x^3 + b$ (of $j$-invariant $0$) over a finite field $\mathbb{F}_q$ such that $\sqrt{b}, \sqrt[3]{b} \not\in \mathbb{F}_q$. Also, for any $n \in \mathbb{...
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Auto-equivalences of non-trivial components of derived category of $X_{18}$
Let $X:=X_{18}$ be an index one smooth prime Fano threefold of degree 18.
Consider its semi-orthogonal decomposition: $D^b(X)=\langle\mathcal{O}_X,\mathcal{E}^{\vee},\mathcal{A}_X\rangle=\langle\...
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Generalization of torsion points on Jacobian of genus 2 over finite fields (with respect to the theta divisor)
Let $J(C)$ be the jacobian of a hyperelliptic curve $C$ of genus 2 defined over finite field $\mathbb{F}_q$. Let $\Theta$ be the image of the curve on the Jacobian under the embedding $P \mapsto P - \...
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Degenerations of hyperelliptic coverings
Take six distinct points $p_1,\dots,p_6\in\mathbb{P}^1$ and consider the double covering $f:C\rightarrow \mathbb{P}^1$ ramified over $p_1,\dots,p_6\in\mathbb{P}^1$. Then $C$ is a smooth curve of genus ...
user125056
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The unit root subspace of a genus-2 odd degree hyperelliptic curve of semistable reduction
Let $K$ be a finite extension of $\mathbb{Q}_p$. If $A_K$ is a semistable Abelian variety over $K$, then we have a Frobenius endomorphism on $H_{dR}^1(A_K)$, whose definition depends on a choice of a ...
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rational points of a hyperelliptic curve of genus 3
Let $K=\mathbb{Q}(\sqrt{-1}).$ I have the following hyperelliptic curve of genus 3:
$$ C : y^2 = (x^2-x+1)(x^6+x^5-6x^4 -3x^3+14x^2-7x+1) $$
I want to find $C(K)$. My first attempt was to compute the ...
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Elliptic factors in the Jacobian and zeta function
Consider a hyperelliptic curve $\mathcal{C}$ over $\mathbb{Q}$ and its Jacobian $J(\mathcal{C})$. Assume that $J(\mathcal{C})$ admits an elliptic factor $\mathcal{E}$.
For almost all primes, we can ...
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Hyperelliptic equation on a function field
Let us consider a hyperelliptic equation
$$Y^2=A_nX^n+A_{n-1}X^{n-1}+\dots+A_0$$
where $A_i\in\mathbb{C}[z]$. I am interested in rational solutions $X,Y\in\mathbb{C}(z)$ when genus is $\geq 2$ and ...
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Concerning the omnipresence of hyper-elliptic curves in the construction of examples
Vague rambling: I hate asking these types of questions, but I feel that I would benefit immensely from hearing some discussion of the use of hyperelliptic curves in constructing certain examples. What ...
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Degree of morphisms and isogenies
$\renewcommand{\J}{\mathrm{Jac}} \renewcommand{\F}{\mathbb{F}}$
I am reading this paper by B. Gross, and there is something I don't understand on p. 945. Here is the context: fix a prime $p \equiv 3 \...
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Is the Cassels "$x - \theta$" map algebraic in some sense?
Setup: Let $k$ be a field of characteristic $0$, let $f(x) \in k[x]$ be a monic separable polynomial of degree $n \geq 4$, and let $\theta$ denote the image of $x$ under the map $k[x] \to K_f := k[x]/(...
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Computing the class group of a quadratic function field
I am asking for a reference in which I can find tools to answer questions like the following: Let $K=\mathbb{F}_q(X)$ be a rational function field over the finite field with $q$ elements. Let $E/K$ be ...
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An isogeny between Jacobians of hyperelliptic curves
Let $\mathbf{F}_q$ be a finite field of odd characteristic. Let $X_t$ be the hyperelliptic curve over $\mathbf{F}_{q^2}(t)$ with affine equation
$$y^2 = \left((x^{(q+1)/2}-(x-1)^{(q+1)/2})^2 - t\...
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What is the quotient $E \!\times\! E^\prime / G$?
Consider a finite field $\mathbb{F}_p$ such that $p \equiv 1 \ (\mathrm{mod} \ 3)$, $p \equiv 3 \ (\mathrm{mod} \ 4)$, $\mathbb{F}_{p^2}$-isomorphic elliptic curves (of $j$-invariant $0$)
$$
E\!:y_1^2 ...
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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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Identifying elements in the kernel of an explicit endomorphism of a Jacobian variety
I hope this question fits here.
Let $H/k$ be a genus $2$ curve and $J$ its Jacobian variety. Since $J(k)\cong \text{Pic}^0(H)(k)$ we have that its generic point looks like $[(x_1,y_1)+(x_2,y_2)-2\...
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$p$ -adic periods of modular curves X_0(71)
I have seen in some papers computation of $p$-adic periods of modular curves $X_0(N)$. Can somebody please explain to me what are the possible applications of such computations?
as a concrete ...
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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 ...
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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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No rational points on $x^n+a=y^2$ for all $n>4$"?
Is there rational (or better integer) $a$ such that for all $n>4$,$x^n+a=y^2$
has no rational points?
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Some curves on the Jacobian of a genus $2$ curve and their image under certain maps (char $p$)
I hope this question belongs here. The situation in this question is quite particular and specific.
I am trying to weak some of theory to measure the degree of some function on the Jacobian of a ...
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Is it true that every mapping class in $\mathrm{Mod}(\Sigma_3)$ commutes with some hyperelliptic involution?
Two questions.
First, let $\Sigma_3$ be the closed genus 3 surface and let $\rm Mod(\Sigma_3)$ be its mapping class group. Is it true that for any mapping class $g\in\rm Mod(\Sigma_3)$ there is some ...
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Genus=2 theta functions, Arnold's relation, and KZ connection
Let $C_5:=\{{(z_1 \dots, z_5) \in (\mathbb{C})^5 | z_i \neq z_j \forall i\neq j }\}$ be the configuration space of five distinct ordered points in $\mathbb{C}$. Arnold showed that the holomorphic one ...
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Intersection number of divisors on abelian surfaces and its invariance under translation by 2-Torsion points
I am reading Fulton's but I cannot find a useful result for my problem that seems to be something well known.
Let $\mathcal{J}$ be the Jacobian of a hyperelliptic curve of genus $2$. Consider $D_1,...
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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)
$$
over $\mathbb{A}^1_t$. Over a generic point the fiber is a genus three curve where one of the genera comes ...
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Is there a finite number of supersingular genus 2 curves?
Consider an algebraically closed field $k$ of prime characteristic. It is widely known that up to $k$-isomorphism there is a finite number of supersingular elliptic curves over $k$. Let $C$ be a ...
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How to describe the subspace of invariants under the Rosati involution?
Consider the Jacobian $J_C$ of hyperelliptic curve
$$C\!: y^2 = x^5 + a$$
over a finite field $\mathbb{F}_p$, where $a \in \mathbb{F}_p^*$, $p \equiv 2 \ (\mathrm{mod} \ 5)$, $p > 2$. Let $\pi \...
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Self intersection of theta divisor
I hope my question is not too basic here.
I have a very specific setting and I am trying to not use "big theorems" to prove a well known fact algebraically. I hope someone can give me a hint.
Let $J/...
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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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Degree of irrationality and hyperelliptic curves
For a variety $V$ of dimension $n$, let $Irr(V)$ denote the minimal degree of a dominant rational map $V\to \mathbb{P}^n$.
Suppose that a curve $X$ admits a dominant map from a variety $V$ with $...
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Birational map from even to odd degree curve. What is the image of one of infinity points?
Suppose I have a curve $C_1$ determined by $C_1: y^2 = (x+a)(f_{2g+1} x^{2g+1} + \dots + f_0)$. It has even degree polynomial of $x$ on the right side. I want to consider its image under birational ...
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Restricted degree function of an endomorphism of a Jacobian to its theta divisor for genus 2 curves
I hope my question is not too vague or basic to be here.
I have been constructing a setting to count points on a curve, but I am stucked solving one part of my problem for some time. Now I would like ...
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Linear systems and 2-torsion shifts on hyperelliptic curves
Let $C$ be a hyperelliptic curve of genus $g$ and let $D$ be a divisor on $C$ of degree $g+1$. Assume that the linear system $|D|$ is base-point-free. Now add a $2$-torsion point $[E]$ to $D$. I would ...
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About the characteristic polynomial of Frobenius of the Jacobian of a genus 2 hyperelliptic curve
I was looking for some information related to the values of the characteristic polynomial $\chi(t)$ of the Frobenius of a Jacobian of a hyperelliptic curve $C$ of genus 2 over $\mathbb{F}_q$ and in ...
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Curves of higher genus
I saw the question:
Abelian varieties with CM
and though I know that there are rare CM elliptic curves, I wonder
what kind of curves with higher genus have the CM Jacobians?
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How order of divisor with support at infinity is changed at reduction?
Jing Yu in his paper "On Arithmetic Of Hyperelliptic Curves" on page 5 asserts the following
The most interesting case is certainly the case $k = \mathbb Q$ and $D \in \mathbb Z[t]$.
To decide ...
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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 ...
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