Questions tagged [hyperelliptic-curves]
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76
questions
17
votes
1
answer
863
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Reference Request: Conductors of Twists of Hyperelliptic Curves
It is my understanding that if I twist a hyperelliptic curve of genus 2 whose Jacobian has conductor $N$ by a prime $p$ with $p\nmid N$, that the conductor of the Jacobian of the twist is expected to ...
16
votes
1
answer
732
views
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 $$w^n = \...
12
votes
1
answer
2k
views
rational points of a hyperelliptic curve
I have the following hyperelliptic curve of genus $2$:
$$
y^2 = 561 x^6 - 41904 x^5 + 627264 x^4 + 11860992 x^3 - 197074944 x^2 + 124416^2
$$
I need to find all the rational points on this curve. ...
12
votes
2
answers
721
views
An isogeny between Jacobians of hyperelliptic curves
Let $\mathbf{F}_q$ be a finite field of odd characteristic. Let $X_t$ be the hyperelliptic curve over $\mathbf{F}_{q^2}(t)$ with affine equation
$$y^2 = \left((x^{(q+1)/2}-(x-1)^{(q+1)/2})^2 - t\...
11
votes
2
answers
604
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 ...
9
votes
1
answer
620
views
Motives over finite field not generated by hyperelliptic curves
So the question is that, over a finite field, does there exist an abelian variety $A$ for which there does not exist a generically one-to-one morphism from a hyperelliptic curve $C$ to $A$.
p.s. A ...
8
votes
5
answers
3k
views
Good lecture notes/books on Jacobian of hyperelliptic curve
I want to understand what the Jacobian variety is from an algebraic (or arithmetic?) perspective.
I want to know:
What is the definition of the Jacobian?
Widely know facts about it.
Why the Jacobian ...
8
votes
0
answers
293
views
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 ...
7
votes
1
answer
308
views
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 ...
7
votes
2
answers
1k
views
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 ...
7
votes
1
answer
450
views
Intermediate Jacobians of intersections of two quadrics
Let
$$X: \quad Q_1(x)=Q_2(x) = 0 \quad \subset \mathbb{P}^{2n+1},$$
be a smooth complete intersection of two quadrics of odd dimension over a field $k$, not of characteristic $2$. Let $J(X)$ denote ...
7
votes
1
answer
264
views
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 ...
7
votes
0
answers
146
views
$p$ -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 ...
6
votes
3
answers
588
views
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 ...
6
votes
1
answer
914
views
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 $...
6
votes
0
answers
202
views
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 ...
6
votes
0
answers
279
views
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 ...
5
votes
1
answer
646
views
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 ...
5
votes
1
answer
247
views
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 $...
5
votes
1
answer
185
views
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\...
5
votes
0
answers
233
views
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?
4
votes
3
answers
563
views
Reference for hyperelliptic curves
I was reading a paper the other day that said that all automorphisms of a hyperelliptic curve are liftings of automorphisms of $\mathbb{P}^1$ operating on the set of branch points.
Can someone point ...
4
votes
1
answer
237
views
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}(...
4
votes
1
answer
442
views
Examples of hyperelliptic curves with hyperelliptic quotients that have more automorphisms
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\...
4
votes
1
answer
484
views
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,...
4
votes
1
answer
684
views
When does a hyperelliptic Riemann surface admit a map of degree 3
Let $X$ be a hyperelliptic curve of genus $g>1$.
For which $g$ does $X$ admit a map $X\to \mathbb P^1$ of degree $3$?
I think a genus two curve $X$ admits a map of degree $3$.
Proof: Pick $P$ ...
4
votes
1
answer
296
views
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\...
4
votes
0
answers
84
views
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 ...
4
votes
0
answers
195
views
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 ...
4
votes
0
answers
112
views
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 \...
4
votes
0
answers
199
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}...
3
votes
2
answers
374
views
Non trivial family of hyperelliptic curves
Let $X$ ba a smooth hyperelliptic curve of genus $g$, and let $f:X\rightarrow X$ be the hyperelliptic involution. Consider a $K3$ surface $S$ with an involution $g$ without fixed points. The quotient $...
3
votes
1
answer
207
views
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 ...
3
votes
1
answer
677
views
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 ...
3
votes
1
answer
313
views
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 - \...
3
votes
1
answer
213
views
Hyperelliptic integrals
I am learning about hyperelliptic curves and hyperelliptic integrals. I encountered some problems when reading the book by Gesztesy and Holden (F. Gesztesy, H. Holden, Soliton Equations and Their ...
3
votes
1
answer
918
views
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 ...
3
votes
0
answers
137
views
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, ...
3
votes
0
answers
186
views
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]/(...
3
votes
0
answers
167
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 ...
3
votes
0
answers
113
views
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 ...
3
votes
0
answers
593
views
Hyperelliptic Curve [closed]
Consider the curve given by $z^{2g-2}y^2=\displaystyle\prod_{i=1}^{2g}(x-a_iz)$. This is a hyperelliptic curve and has genus $g-1$. At the same time it is a curve defined by an equation of $d=2g$ and ...
2
votes
2
answers
646
views
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?
2
votes
1
answer
146
views
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 ...
2
votes
1
answer
369
views
Function field of the Jacobian of genus 2 curve over $\mathbb{F}_q$
I have been trying to build the function field of the jacobian of a genus 2 smooth curve over a finite field, but I am having problems making it explicit, I need to work with another curve with points ...
2
votes
1
answer
201
views
What is the value of this hyperelliptic Hodge-type integral?
Consider the moduli space
$$
\overline{M}_{0,4}(B\mathbb{Z}/2)
$$
This has virtual (and real) dimension one. In a certain sense this moduli space paramaterizes "genus 1 hyperelliptic curves"; that is, ...
2
votes
4
answers
544
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 ...
2
votes
1
answer
381
views
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 \...
2
votes
1
answer
169
views
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 ...
2
votes
1
answer
189
views
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 ...