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Eduardo R. Duarte
Eduardo R. Duarte
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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 some sources it says that it provides information of #$\mathcal{J}_C(\mathbb{F}_{q^k})$ for every $k$ , I just proved that $\chi(-1)=$ #$\mathcal{J}_{C^{TW}}(\mathbb{F}_q)$ which is the jacobian of the quadratic twist of the hyperelliptic curve, so , the value $\chi(-1)$ is not giving me information about the jacobian over $\mathbb{F}_{q^k}$, even that I know that the twist of $C$ and $C^{TW}$ are isomorphic over $\mathbb{F}_{q^2}$.

One question is, where I can find more information about the values of $\chi$.

I am interested in the value of $\chi(2)$, what I am doing is considering the degree of the map $\Phi+[\mathcal{n}]$ (where $[1],\Phi\in End_{\mathbb{F}_q}(\mathcal{J}_{C})$ are the identity and Frobenius, respectively), it is known that $\Phi-[1]$ is a separable map, so # $ker(\Phi+[-1])=$ # $\mathcal{J}_C(\mathbb{F}_q)$ , in fact $\chi(n)=\partial (\Phi+[n])$,

what can be said about $\chi(2)$?, or about $\chi(n)$?

I am working with an specific example to gain some intuition, in fact is $C_{\mathbb{F}_{101}}$ given by $y^2 = x^5 + x + 1$ where the characteristic polynomial of the Frobenius is given by $\chi(t) = L^C_{\mathbb{F}_q}(1/t)$$\chi(t) = t^{2g}L^C_{\mathbb{F}_q}(1/t)$ , which in this case is: $t^4 + 2t^3 + 26t^2 + 202t + 10201$, and $\chi(2)=10741$.

I also saw in the slide 22 of this presentation that #$\mathcal{J}_C(\mathbb{F}_p)=\frac{1}{2}$#$C(\mathbb{F}_{p^2})+\frac{1}{2}$#$C(\mathbb{F}_p)-p$ which is false. (see magma example below)

http://www.skidmore.edu/fq12/uploads/Eisentraeger.pdf

Example of magma, showing that the last equality in slide 22 is false.


> p := 101;
> Fp := GF(p);
> _ := PolynomialRing(Fp);
> f := x^5 + x + 1;
> CFp := HyperellipticCurve(f);
> CFp2 := BaseExtend(CFp,GF(p^2));
> J := Jacobian(CFp);
> #J;   
10432
> ((#CFp+#CFp2)/2)-p;
5076
> #CFp;#CFp2;
104
10250
> CFp;CFp2;J;
Hyperelliptic Curve defined by y^2 = x^5 + x + 1 over GF(101)
Hyperelliptic Curve defined by y^2 = x^5 + x + 1 over GF(101^2)
Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + x + 1 over GF(101)
>

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 some sources it says that it provides information of #$\mathcal{J}_C(\mathbb{F}_{q^k})$ for every $k$ , I just proved that $\chi(-1)=$ #$\mathcal{J}_{C^{TW}}(\mathbb{F}_q)$ which is the jacobian of the quadratic twist of the hyperelliptic curve, so , the value $\chi(-1)$ is not giving me information about the jacobian over $\mathbb{F}_{q^k}$, even that I know that the twist of $C$ and $C^{TW}$ are isomorphic over $\mathbb{F}_{q^2}$.

One question is, where I can find more information about the values of $\chi$.

I am interested in the value of $\chi(2)$, what I am doing is considering the degree of the map $\Phi+[\mathcal{n}]$ (where $[1],\Phi\in End_{\mathbb{F}_q}(\mathcal{J}_{C})$ are the identity and Frobenius, respectively), it is known that $\Phi-[1]$ is a separable map, so # $ker(\Phi+[-1])=$ # $\mathcal{J}_C(\mathbb{F}_q)$ , in fact $\chi(n)=\partial (\Phi+[n])$,

what can be said about $\chi(2)$?, or about $\chi(n)$?

I am working with an specific example to gain some intuition, in fact is $C_{\mathbb{F}_{101}}$ given by $y^2 = x^5 + x + 1$ where the characteristic polynomial of the Frobenius is given by $\chi(t) = L^C_{\mathbb{F}_q}(1/t)$ , which in this case is: $t^4 + 2t^3 + 26t^2 + 202t + 10201$, and $\chi(2)=10741$.

I also saw in the slide 22 of this presentation that #$\mathcal{J}_C(\mathbb{F}_p)=\frac{1}{2}$#$C(\mathbb{F}_{p^2})+\frac{1}{2}$#$C(\mathbb{F}_p)-p$ which is false. (see magma example below)

http://www.skidmore.edu/fq12/uploads/Eisentraeger.pdf

Example of magma, showing that the last equality in slide 22 is false.


> p := 101;
> Fp := GF(p);
> _ := PolynomialRing(Fp);
> f := x^5 + x + 1;
> CFp := HyperellipticCurve(f);
> CFp2 := BaseExtend(CFp,GF(p^2));
> J := Jacobian(CFp);
> #J;   
10432
> ((#CFp+#CFp2)/2)-p;
5076
> #CFp;#CFp2;
104
10250
> CFp;CFp2;J;
Hyperelliptic Curve defined by y^2 = x^5 + x + 1 over GF(101)
Hyperelliptic Curve defined by y^2 = x^5 + x + 1 over GF(101^2)
Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + x + 1 over GF(101)
>

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 some sources it says that it provides information of #$\mathcal{J}_C(\mathbb{F}_{q^k})$ for every $k$ , I just proved that $\chi(-1)=$ #$\mathcal{J}_{C^{TW}}(\mathbb{F}_q)$ which is the jacobian of the quadratic twist of the hyperelliptic curve, so , the value $\chi(-1)$ is not giving me information about the jacobian over $\mathbb{F}_{q^k}$, even that I know that the twist of $C$ and $C^{TW}$ are isomorphic over $\mathbb{F}_{q^2}$.

One question is, where I can find more information about the values of $\chi$.

I am interested in the value of $\chi(2)$, what I am doing is considering the degree of the map $\Phi+[\mathcal{n}]$ (where $[1],\Phi\in End_{\mathbb{F}_q}(\mathcal{J}_{C})$ are the identity and Frobenius, respectively), it is known that $\Phi-[1]$ is a separable map, so # $ker(\Phi+[-1])=$ # $\mathcal{J}_C(\mathbb{F}_q)$ , in fact $\chi(n)=\partial (\Phi+[n])$,

what can be said about $\chi(2)$?, or about $\chi(n)$?

I am working with an specific example to gain some intuition, in fact is $C_{\mathbb{F}_{101}}$ given by $y^2 = x^5 + x + 1$ where the characteristic polynomial of the Frobenius is given by $\chi(t) = t^{2g}L^C_{\mathbb{F}_q}(1/t)$ , which in this case is: $t^4 + 2t^3 + 26t^2 + 202t + 10201$, and $\chi(2)=10741$.

I also saw in the slide 22 of this presentation that #$\mathcal{J}_C(\mathbb{F}_p)=\frac{1}{2}$#$C(\mathbb{F}_{p^2})+\frac{1}{2}$#$C(\mathbb{F}_p)-p$ which is false. (see magma example below)

http://www.skidmore.edu/fq12/uploads/Eisentraeger.pdf

Example of magma, showing that the last equality in slide 22 is false.


> p := 101;
> Fp := GF(p);
> _ := PolynomialRing(Fp);
> f := x^5 + x + 1;
> CFp := HyperellipticCurve(f);
> CFp2 := BaseExtend(CFp,GF(p^2));
> J := Jacobian(CFp);
> #J;   
10432
> ((#CFp+#CFp2)/2)-p;
5076
> #CFp;#CFp2;
104
10250
> CFp;CFp2;J;
Hyperelliptic Curve defined by y^2 = x^5 + x + 1 over GF(101)
Hyperelliptic Curve defined by y^2 = x^5 + x + 1 over GF(101^2)
Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + x + 1 over GF(101)
>
Source Link
Eduardo R. Duarte
Eduardo R. Duarte
  • 795
  • 3
  • 9

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 some sources it says that it provides information of #$\mathcal{J}_C(\mathbb{F}_{q^k})$ for every $k$ , I just proved that $\chi(-1)=$ #$\mathcal{J}_{C^{TW}}(\mathbb{F}_q)$ which is the jacobian of the quadratic twist of the hyperelliptic curve, so , the value $\chi(-1)$ is not giving me information about the jacobian over $\mathbb{F}_{q^k}$, even that I know that the twist of $C$ and $C^{TW}$ are isomorphic over $\mathbb{F}_{q^2}$.

One question is, where I can find more information about the values of $\chi$.

I am interested in the value of $\chi(2)$, what I am doing is considering the degree of the map $\Phi+[\mathcal{n}]$ (where $[1],\Phi\in End_{\mathbb{F}_q}(\mathcal{J}_{C})$ are the identity and Frobenius, respectively), it is known that $\Phi-[1]$ is a separable map, so # $ker(\Phi+[-1])=$ # $\mathcal{J}_C(\mathbb{F}_q)$ , in fact $\chi(n)=\partial (\Phi+[n])$,

what can be said about $\chi(2)$?, or about $\chi(n)$?

I am working with an specific example to gain some intuition, in fact is $C_{\mathbb{F}_{101}}$ given by $y^2 = x^5 + x + 1$ where the characteristic polynomial of the Frobenius is given by $\chi(t) = L^C_{\mathbb{F}_q}(1/t)$ , which in this case is: $t^4 + 2t^3 + 26t^2 + 202t + 10201$, and $\chi(2)=10741$.

I also saw in the slide 22 of this presentation that #$\mathcal{J}_C(\mathbb{F}_p)=\frac{1}{2}$#$C(\mathbb{F}_{p^2})+\frac{1}{2}$#$C(\mathbb{F}_p)-p$ which is false. (see magma example below)

http://www.skidmore.edu/fq12/uploads/Eisentraeger.pdf

Example of magma, showing that the last equality in slide 22 is false.


> p := 101;
> Fp := GF(p);
> _ := PolynomialRing(Fp);
> f := x^5 + x + 1;
> CFp := HyperellipticCurve(f);
> CFp2 := BaseExtend(CFp,GF(p^2));
> J := Jacobian(CFp);
> #J;   
10432
> ((#CFp+#CFp2)/2)-p;
5076
> #CFp;#CFp2;
104
10250
> CFp;CFp2;J;
Hyperelliptic Curve defined by y^2 = x^5 + x + 1 over GF(101)
Hyperelliptic Curve defined by y^2 = x^5 + x + 1 over GF(101^2)
Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + x + 1 over GF(101)
>