18
votes

Accepted

### rational points of a hyperelliptic curve

By now there is a fairly rich literature on computing the set of rational points on curves of higher genus, see for example my survey paper on "Rational points on curves".
What one can do for your ...

18
votes

Accepted

### Rational perfect power values of $y(y+1)$

There are no such solutions. Let $x=a/b$ and $y=c/d$ be reduced fractions. Then
$a^n/b^n=(c(c+d))/d^2$ and since both sides are reduced fractions we get that
$a^n=c(c+d)$ and $b^n=d^2$. From the ...

16
votes

### rational points of a hyperelliptic curve of genus 3

It turns out that $C(K) = C(\mathbb Q) = \{\infty_+, \infty_-, (0,1), (0,-1), (1,1), (1,-1)\}$.
To see this, consider a point $P \in C(K)$ and write $\bar{P}$ for its image under the nontrivial ...

14
votes

Accepted

### From a physicist: How do I show certain superelliptic curves are also hyperelliptic?

This curve is not hyperelliptic unless $n=2$ or $N=2$.
First, note that it is more convenient to write the curve as
$$ w^n = \frac{ \prod_{\alpha=1}^N (z- u_\alpha )} {\prod_{\alpha=1}^N (z- v_\alpha)}...

13
votes

### Good lecture notes/books on Jacobian of hyperelliptic curve

Over $\mathbb C$, there's also Mumford's beautiful monograph Curves and their Jacobians, The University of Michigan Press, Ann Arbor, Mich., 1975. But a monograph covering all of the topics in your ...

10
votes

### Easiest example where field of definition is not field of moduli

EDIT: This answer is incorrect, for the reason indicated in the other answer; it should be consulted for a correct curve.
Here is a recipe for constructing some examples.
Suppose that $f$ is a ...

10
votes

Accepted

### Non trivial family of hyperelliptic curves

If it was, $Y:=(X\times S)/(f\times g)$ would be isomorphic to $X\times F$. One way to see this is not the case is to look at 3-forms: $X\times F$ has no nonzero holomorphic 3-forms (because $H^0(F,\...

7
votes

### About the characteristic polynomial of Frobenius of the Jacobian of a genus 2 hyperelliptic curve

The equation for $\# J_C(\mathbb F_p)$ that you quote contains a typo: they must have meant that $\# J_C(\mathbb F_p) = \frac 1 2 \# C(\mathbb F_{p^2}) + \frac 1 2 \# C(\mathbb F_p)^2 - p$, which is ...

7
votes

Accepted

### Hyperelliptic curve of genus 2 over R

Basically, the problem is that there are really five intersection points with your line (two of them are complex non-real). If you could make a group in the same way as for elliptic curves, then this ...

7
votes

### Is it true that every mapping class in $\mathrm{Mod}(\Sigma_3)$ commutes with some hyperelliptic involution?

No. After the Nielsen realization theorem, this follows from the classification of automorphism groups of genus three Riemann surfaces, which can be found in many places. For example, the cyclic group ...

6
votes

### Intermediate Jacobians of intersections of two quadrics

This is not an answer, but this is too long for a comment.
The hyperelliptic curve (let me call its $C$) associated to an intersection of quadrics $X$ has an intrinsic meaning as the moduli space of ...

6
votes

Accepted

### Curves of higher genus

I am not sure whether this answers your question: it is a conjecture of Coleman that for a fixed genus $g$ sufficiently high, there should be only finitely many CM Jacobians of genus $g$. In fact ...

6
votes

Accepted

### Degenerations of hyperelliptic coverings

If $p_1 = p_2 \ne p_3 = p_4 \ne p_5 \ne p_6$ then the normalization of the double cover branched at the divisor $D = \sum_{i=1}^6 p_i$ is a smooth irreducible rational curve. If also $p_5$ and $p_6$ ...

5
votes

### Rational functions on hyperelliptic Riemann surface

Yes (the answer was given, then deleted, by Francesco Polizzi). If $D$ and $D'$ are the divisors of zeroes (resp. poles) of a rational function, the linear system $|D|$ has dimension $r\geq 1$ and is ...

5
votes

Accepted

### Examples of hyperelliptic curves with hyperelliptic quotients that have more automorphisms

Sure: let $Y$ be your favorite hyperelliptic curve $u^2=f(t)$
with many automorphisms, and let $X$ be the curve $u^2=f(t(s))$
for some "random" rational function $f$ of degree at least $2$.
For ...

5
votes

### Good lecture notes/books on Jacobian of hyperelliptic curve

FWIW, for a quick intro, here are notes from Day 1 of my course:
8320 Spring 2010 Day one: Introduction to Riemann Surfaces
We will describe how Riemann used ...

5
votes

### Good lecture notes/books on Jacobian of hyperelliptic curve

This is a really big topic. So I'll treat it entirely as reference request.
The Jacobian of a smooth projective curve $X$ of genus $g$ is an abelian variety of dimension $g$ whose group of points is ...

5
votes

Accepted

### Degree of irrationality and hyperelliptic curves

Yes, because then there is a nonconstant map from projective space to the symmetric square of $C$.

5
votes

### Elliptic factors in the Jacobian and zeta function

This is an incomplete answer to the second question (Jackson has indicated the answer to the first question in the comments).
If you ask the same question over a general number field $F$, then you ...

5
votes

Accepted

### Finding the $K=\mathbb{Q}(\sqrt{6})$-rational points on the twist of $X_{0}(26)$

I used Magma to point search on $C/K$ up to a height of $1000$ and it appears that $C(K) = \emptyset$. If that's true, then one can probably use the Mordell-Weil sieve to prove it. Here's a bit more ...

4
votes

### What is the value of this hyperelliptic Hodge-type integral?

You are right that it's equivalent to compute the degree of the forgetful map. The degree is $3!$, since it's the quotient by $S_3$ permuting the last three markings. If you divide by $S_4$ you get ...

4
votes

### Easiest example where field of definition is not field of moduli

I believe that the other answer to this question is incorrect (I don't have the reputation to comment on it). The curve
$$y^2=(x^2-1)(x^2-2)(x-i)$$
is in fact defined over $\mathbb{R}$. To see this ...

4
votes

Accepted

### Calculate reduction of Jacobian of hyperelliptic curve

The relevant information can be obtained from a regular model of the curve over ${\mathbb Z}_p$. Such a model can be computed by repeatedly blowing up points or components of the special fiber that ...

4
votes

Accepted

### Function field of the Jacobian of genus 2 curve over $\mathbb{F}_q$

You get an open subset of the Jacobian by looking at points "in general
position", i.e., points represented by divisors of the form $(P)+(P')-2(\infty)$,
where $\infty$ denotes the point at infinity, ...

4
votes

Accepted

### Reference Request: Conductors of Twists of Hyperelliptic Curves

(This answer is Community Wiki and is extracted from comments by user eric, who has declined to post them as an answer. The CW is to invite others to contribute, especially to provide suitable ...

Community wiki

4
votes

### Is the Jacobian isogenous over $\mathbb{F}_p$ to the direct product of the elliptic curves?

If $$y^2 = (x^3+b) (x^3-b) = x^6-b^2$$ then $$(y)^2 = (x^2)^3 - (b^2)$$ and $$(y x^{-3} b^{-1} )^2 = (- x^{-2})^3 + (b^{-2}) $$ giving two maps to elliptic curves, but not the elliptic curves you ...

4
votes

### An isogeny between Jacobians of hyperelliptic curves

In the most easy case $q=3$, the curve $X_t$ is bielliptic (the bielliptic involution given by $x\mapsto 1-x$), and the Jacobian of $X_t$ is then $(2,2)$-isogenous to the product
$E_{1,t}\times E_{2,...

4
votes

Accepted

### Generalization of torsion points on Jacobian of genus 2 over finite fields (with respect to the theta divisor)

The set $J(C)_{\Theta}[n]$ has the structure of a smooth irreducible algebraic curve, and the restriction of $J(C)\xrightarrow{\times n } J(C)$ to $C$ defines a morphism $J(C)_{\Theta}[n]\rightarrow C$...

3
votes

### Curves of higher genus

It should be noted that Murabayashi determined that there should be finitely many (whose moduli lie in the rational numbers) for $g=2$ over the complex numbers and (mostly) explicitly determined them.
...

3
votes

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

I think the answer is no. Here is an argument.
Degenerate your $8$ points so that there are $2$ $5$-tuples that lie in $3$-planes $A,B$. The degree $3$ rational curves through the first $5$-tuple ...

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