Supersingular genus-2 curves over fields of characteristic 3.

*(English)*Zbl 1166.11020
Lauter, Kristin E. (ed.) et al., Computational arithmetic geometry. AMS special session, San Francisco, CA, USA, April 29–30, 2006. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4320-8/pbk). Contemporary Mathematics 463, 49-69 (2008).

The main purpose of the article is to complete the result of E. W. Howe, E. Nart and the reviewer [Ann. Inst. Fourier 59, No. 1, 239–289 (2009; Zbl 1236.11058)] by giving the Weil polynomials for supersingular genus \(2\) curves over the finite fields \(\mathbb{F}_{q}\) with \(q=3^d\). The result is the following: if \(d\) is odd the Weil polynomial belongs to the following list

1. \((x^2+q)(x^2-sx+q)\) for all \(s \in \{\pm \sqrt{3q}\}\);

2. \((x^2+q)^2\), if \(q>3\);

3. \(x^4+q\);

4. \(x^4+qx^2+q^2\);

5. \(x^4-2qx^2+q^2\) if \(q>3\).

And if \(d\) is even it belongs to the list

1) \((x^2-2sx+q)(x^2+sx+q)\) for all \(s \in \{\pm \sqrt{q}\}\);

2) \((x^2-sx+q)^2\) for all \(s \in \{0,\pm \sqrt{q}\}\);

3) \((x^2-2sx+q)^2\) for all \(s \in \{\pm \sqrt{q}\}\), if \(q>9\);

4) \(x^4+q^2\);

5) \(x^4-sx^3+qx^2-sqx+q^2\) for all \(s\in \{\pm \sqrt{q}\}\).

The Weil polynomials of abelian surfaces which cannot be obtained are treated using various arguments. For positive results, note that for each class, an explicit curve can be constructed. This is done by using the first part of the paper which gives a beautiful analysis of the coarse moduli space \(\mathcal{A}\) of triples \((C,E,\phi)\) where \(C\) is a supersingular genus \(2\) curve over a field of characteristic \(3\), \(E\) the elliptic curve with \(j\)-invariant \(0\) and \(\phi :C \to E\) a degree \(3\) map. The author shows (Th. 2.1) that \(C\) is supersingular if and only if its Igusa invariants \(J_2,J_4\) and \(J_8\) are zero and shows that \(C : y^2=x^6+A x^3+Bx+A^2\) represents the invariant \([0:0:A:0:B]\). Two other nice properties are shown : if \(C\) is a genus \(2\) triple cover of a supersingular elliptic curve in characteristic \(3\) then \(C\) is supersingular (Cor.3.3) ; if \(C : y^2=f(x)\) with \(f\) a sextic polynomial is a supersingular genus \(2\) curve such that \(f\) can be written as the product of two cubic factors then \(C\) is a triple cover of a supersingular elliptic curve (Th.3.4).

Finally the author shows that \(\mathcal{A}\) is isomorphic to the affine line with one point removed and is a degree \(20\) cover of the moduli space of supersingular genus \(2\) curves. On the other hand, the author shows that \(\mathcal{A}\) is also isomorphic to the moduli space of pairs \((C,G)\) where \(C\) is supersingular and \(G\) is an order \(4\) subgroup of \(\mathrm{Jac}(C)[2]\) that is not isotropic with respect to the Weil pairing.

For the entire collection see [Zbl 1143.11002].

1. \((x^2+q)(x^2-sx+q)\) for all \(s \in \{\pm \sqrt{3q}\}\);

2. \((x^2+q)^2\), if \(q>3\);

3. \(x^4+q\);

4. \(x^4+qx^2+q^2\);

5. \(x^4-2qx^2+q^2\) if \(q>3\).

And if \(d\) is even it belongs to the list

1) \((x^2-2sx+q)(x^2+sx+q)\) for all \(s \in \{\pm \sqrt{q}\}\);

2) \((x^2-sx+q)^2\) for all \(s \in \{0,\pm \sqrt{q}\}\);

3) \((x^2-2sx+q)^2\) for all \(s \in \{\pm \sqrt{q}\}\), if \(q>9\);

4) \(x^4+q^2\);

5) \(x^4-sx^3+qx^2-sqx+q^2\) for all \(s\in \{\pm \sqrt{q}\}\).

The Weil polynomials of abelian surfaces which cannot be obtained are treated using various arguments. For positive results, note that for each class, an explicit curve can be constructed. This is done by using the first part of the paper which gives a beautiful analysis of the coarse moduli space \(\mathcal{A}\) of triples \((C,E,\phi)\) where \(C\) is a supersingular genus \(2\) curve over a field of characteristic \(3\), \(E\) the elliptic curve with \(j\)-invariant \(0\) and \(\phi :C \to E\) a degree \(3\) map. The author shows (Th. 2.1) that \(C\) is supersingular if and only if its Igusa invariants \(J_2,J_4\) and \(J_8\) are zero and shows that \(C : y^2=x^6+A x^3+Bx+A^2\) represents the invariant \([0:0:A:0:B]\). Two other nice properties are shown : if \(C\) is a genus \(2\) triple cover of a supersingular elliptic curve in characteristic \(3\) then \(C\) is supersingular (Cor.3.3) ; if \(C : y^2=f(x)\) with \(f\) a sextic polynomial is a supersingular genus \(2\) curve such that \(f\) can be written as the product of two cubic factors then \(C\) is a triple cover of a supersingular elliptic curve (Th.3.4).

Finally the author shows that \(\mathcal{A}\) is isomorphic to the affine line with one point removed and is a degree \(20\) cover of the moduli space of supersingular genus \(2\) curves. On the other hand, the author shows that \(\mathcal{A}\) is also isomorphic to the moduli space of pairs \((C,G)\) where \(C\) is supersingular and \(G\) is an order \(4\) subgroup of \(\mathrm{Jac}(C)[2]\) that is not isotropic with respect to the Weil pairing.

For the entire collection see [Zbl 1143.11002].

Reviewer: Christophe Ritzenthaler (Marseille)

##### MSC:

11G20 | Curves over finite and local fields |

14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |

14G15 | Finite ground fields in algebraic geometry |