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Questions tagged [hyperelliptic-curves]

The tag has no usage guidance.

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0answers
63 views

Arithmetic genus of intersection

It is common knowledge that using adjunction formula we get genus of intersection of two smooth surfaces in $\mathbb{P}^3$ of degree $d_1$ и $d_2$. But if we take two surfaces $y^2=P_n(x)$ and $z^2+...
5
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1answer
124 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\...
7
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0answers
114 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 ...
7
votes
1answer
211 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 ...
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2answers
273 views

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 ...
5
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0answers
174 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?
3
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1answer
223 views

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 ...
5
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3answers
298 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 ...
8
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0answers
204 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 ...
3
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1answer
222 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,...
3
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0answers
127 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 ...
4
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0answers
143 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 ...
6
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0answers
215 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 ...
3
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0answers
70 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 \...
3
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0answers
225 views

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/...
0
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1answer
151 views

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 ...
5
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1answer
207 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 $...
2
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0answers
97 views

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 ...
2
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1answer
168 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 ...
2
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1answer
148 views

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 ...
3
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1answer
434 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 ...
2
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2answers
584 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?
3
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0answers
97 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
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1answer
833 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 ...
2
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0answers
169 views

Generalized “elliptic integrals”

I am interested in evaluating the following type of integrals. Here we a polynomial $q(x)$ of degree $d \geq 2$ with no non-negative roots. Then is there a name for integrals of the shape $$\int_0^\...
17
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1answer
782 views

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 ...
2
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0answers
51 views

Computations of some character sums/zeta function

I'm currently trying to "compute" the zeta function of some hyperelliptic curves over a finite field (of odd characteristic). Precisely, let $b\in\mathbb{F}_q^\times$ be a non zero element (I also ...
1
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1answer
131 views

Rational functions on hyperelliptic Riemann surface

Let $\mathcal R$ be an hyperelliptic Riemann surface of genus $g\geq 1$. Is it true that the only possible rational functions on $\mathcal R$ with $\leq g$ poles are the liftings of rational functions ...
4
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2answers
385 views

Secant varieties of curves in $\mathbb{P}^4$

My question is motivated by the following simple observations. By a standard dimensions count in $\mathbb{P}^4$ there should not exist neither an hypersurface of degree $3$ with multiplicity $2$ in ...
1
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1answer
256 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 ...
0
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1answer
282 views

Hyperelliptic curve of genus 2 over R

I know that the points of an elliptic curve over $\mathbb{Q}$, $\mathbb{R}$ or other field $K$ form a group, particularly the most common example to explain the naive way is with this curve $y^2=x^3-x$...
4
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1answer
482 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 $P$ ...
2
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1answer
157 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, ...
6
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4answers
2k 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 ...
10
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1answer
1k 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. ...
6
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1answer
294 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 ...
6
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2answers
322 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 $...
4
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1answer
252 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\...
10
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1answer
360 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 ...
2
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4answers
497 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 ...
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0answers
152 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}...
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1answer
102 views

Invertible functions on open subset of hyperelliptic curve

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 ...
6
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1answer
648 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 $...
7
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2answers
763 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 ...
9
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1answer
514 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 ...
2
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1answer
454 views

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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0answers
123 views

Genus 2 hyperelliptic cryptography : typical discriminant and class number

As far as I know, there is no standard yet for cryptography based on the DLP over Jacobians of genus 2 curves. Yet, what can we say about the class number, and the discriminant of the complex ...
3
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0answers
463 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 ...
3
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3answers
402 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 ...