I would like to construct hyperelliptic curves whose Jacobians are isogenous to the square of a supersingular elliptic curve over $\mathbb{F}_{p^2}$
My question is motivated by the following example.
Let $H/\mathbb{F}_{5^2}$ be the hyperelliptic curve given by $y^2 = x^6 + 1$ and $E/\mathbb{F}_{5^2}$ an elliptic curve $y^2 = x^3 + 1$, then we have that $J:=Jac(H)$ has characteristic polynomial of Frobenius given by $(t + 5)^4$.
There is a map $\psi:H\to E$ given by $(x,y)\mapsto (x^2,y)$ and $E$ is supersingular.
If $\phi\in End(J)$ is the $5^2$-Frobenius endomorphism then we have that $\phi = [-5]$
I think this is a direct consequence of $J$ being isogenous to a the square of a supersingular elliptic curve as the Frobenius satifies its characteristic polynomial in the Tate module, but I am not sure if this suffices.
I would like to know if someone can point me out to a characterization of hyperelliptic curves of genus two where the Frobenius in the endomorphism ring of its Jacobian is the $[n]$ map for some $n$.
In fact it will be nice to construct the hyperelliptic curve of genus 2 such that its jacobian is isogenous to the square of a supersingular elliptic curve.
Thanks.
Just for fun, in MAGMA, I have that the curve $H$ is isomorphic to $y^2=\alpha x^5 + \alpha^{13}x$ and the output to check this is using the generic point of $H$ as follows:
Note: I changed the angle brackets by () as I could not find how to show them.
> F(a) := FiniteField(5^2);
> P(x) := PolynomialRing(F);
> f1 := x^6 + 1;
> f := a*x^5 + a^13*x;
> H1 := HyperellipticCurve(f1);
> H := HyperellipticCurve(f);
> IsIsomorphic(H1,H);
true Mapping from: CrvHyp: H1 to CrvHyp: H
with equations :
2*$.1 + 4*$.3
a^5*$.2
a^8*$.1 + a^22*$.3
and inverse
a^22*$.1 + $.3
a^4*$.2
a^20*$.1 + 2*$.3
> FH(X,Y) := FunctionField(H);
> Hext := BaseExtend(H,FH);
> Jext := Jacobian(Hext);
> R(z) := PolynomialRing(FH);
> PtJ := Jext![z-X,Y];
> FrPtJ := Jext![z-X^25,Y^25];
> FrPtJ;
(x + 4*X^25, (4*X^60 + 2*X^56 + 4*X^52 + 2*X^40 + X^36 + 2*X^32 + 4*X^20 + 2*X^16 + 4*X^12)*Y, 1)
> -5*PtJ;
(x + 4*X^25, (4*X^60 + 2*X^56 + 4*X^52 + 2*X^40 + X^36 + 2*X^32 + 4*X^20 + 2*X^16 + 4*X^12)*Y, 1)
> FrPtJ + 5*PtJ;
(1, 0, 0)
> w(t) := PolynomialRing(Rationals());
> t^(2*Genus(H))*Evaluate(Numerator(ZetaFunction(H)),1/t);
t^4 + 20*t^3 + 150*t^2 + 500*t + 625
> Factorization(t^4 + 20*t^3 + 150*t^2 + 500*t + 625);
[
(t + 5, 4)
]
