Hyperelliptic Jacobians and Hasse–Witt Matrix

Recall that a hyperelliptic curve is a smooth projective curve of genus $g \geq 2$ that admits a degree 2 morphism to the projective line $\mathbb{P}^1$. In this post, we will discuss how to compute the Hasse–Witt matrix of a hyperelliptic curve defined over a finite field, which encodes information about the curve's Jacobian. We will also explore some applications of the Hasse–Witt matrix in number theory and algebraic geometry.

Serre Duality

Recall that Serre duality gives a perfect pairing $H^0(X,\Omega^1_X) \times H^1(X,\mathcal O_X) \to k$ defined by $(\omega, \alpha) \mapsto \mathrm{Tr}(\alpha \smile \omega)$, where $\smile$ is the cup product and $\mathrm{Tr}: H^1(X,\Omega^1_X) \to k$ is the trace map. However, the description we shall use here is in terms of residue: $(\omega, \alpha)\mapsto \sum_{P\in |X|}\mathrm{Tr}_{\kappa(P)/k}\mathrm{Res}(\alpha_P \omega)$, where $\alpha_P$ is the local principal part of $\alpha$ at $P$, which we shall explain.

For each divisor $D=\sum_{P}n_P P$, there is a isomorphism of $k$-vector spaces $H^1(X,\mathcal O_X(D))\cong \mathbb{A}_X/(\mathbb{A}_X(D)+\mathcal K(X))$, where $\mathcal K(X)$ embeds diagonally.

Proof.

(Serre Duality)

Arithmetic of Hyperelliptic Curves

Recall a hyperelliptic curve $C/\mathbb F_q$ is a smooth projective curve of genus $\ge 2$ with a degree $2$ map to $\mathbb P^1$. Assume the characteristic of $\mathbb F_q$ is odd, it has affine model $y^2=f(x)$, where $f(x)$ is square free.

The Jacobian $\mathrm{Jac}(C)$ of a smooth projective curve $C$ is a principally polarizaed abelian variety, defined as the identity component the Picard scheme $\mathrm{Pic}^0(C)$, where the Picard functor is the fppf sheafification of the functor $(\mathrm{Sch}^{\mathrm{op}})_{\mathrm{fppf}}\to \mathbf{Ab}$

$$T\mapsto \mathrm{Pic}(C_T)/\pi_*\mathrm{Pic}(T)$$

representable by an abelian variety. A polarization of an abelian variety $A$ is an isogeny $\lambda:A\rightarrow A^{\vee}$, where $A^\vee:=\mathrm{Jac}_{A/k}$ is the dual abelian variety, arising from an ample line bundle in this way: for a line bundle $L$, define

$$\lambda_L:A\to A^\vee\qquad x\mapsto t^*_xL\otimes L^{-1}$$

where $t_x:A\rightarrow A$ is the translation map by $x$. Moreover it is called principal if it is an isomorphism. In the case of a Jacobian $J=\mathrm{Jac}(C)$, there is a principal polarization induced by the ample divisor $\Theta$, defined as the image of $C^{(g-1)}:=C^{g-1}/\mathrm{Sym}_{g-1}\to J$ that sends $D\mapsto \mathcal O_C(D-(g-1)P_0)$, where $P_0$ is a rational point, and $D$ is the divisor representation of an element in $C^{(g-1)}$.

In other words, the Jacobian of $C$ is the moduli space of the degree $0$ line bundles on a $C$. The Neron-Severi group of $C$ is defined to be $\mathrm{Pic}(C)/\mathrm{Pic}^0(C)$. There is a canonical exact sequence.

$$0\to \mathrm{Pic}^0(C)\rightarrow \mathrm{Pic}(C)\rightarrow \mathrm{NS}(C)\rightarrow 0$$

where $\mathrm{NS}(C)=\mathbb Z$ if $k$ is algebraically closed.

For a smooth projective curve $C$ of genus $g$, its Jacobian $J=\mathrm{Jac}(C)$ has dimension $g$.

Proof.

It suffice to show the tangent space $T_0\mathrm{Pic}_{C}=H^1(C, \mathcal O_C)$, then by Serre duality

$$\dim J=\dim T_0 J=\dim T_0 \mathrm{Pic}_C=\dim H^1(C, \mathcal O_C)=\dim H^0(C,\omega_C):=g$$

Let $C_{\varepsilon}:=C\times_{k}k[\varepsilon]/\varepsilon^2$. There is an exact sequence of sheaves

$$0\rightarrow \mathcal O_C\xrightarrow{f\mapsto 1+\varepsilon f} \mathcal O_{C_{\varepsilon}}^\times\xrightarrow{a+\varepsilon b\mapsto a}\mathcal O_{C}^\times\rightarrow 0$$

This induces long exact sequence

$$H^0\left(C, \mathcal{O}_{C_{\varepsilon}}^{\times}\right) \rightarrow H^0\left(C, \mathcal{O}_C^{\times}\right) \rightarrow H^1\left(C, \mathcal{O}_C\right) \longrightarrow H^1\left(C, \mathcal{O}_{C_{\varepsilon}}^{\times}\right)=\mathrm{Pic}(C_{\varepsilon}) \longrightarrow H^1\left(C, \mathcal{O}_C^{\times}\right)=\mathrm{Pic}(C)$$

which gives $H^1(C,\mathcal O_C)=\mathrm{Ker}(\mathrm{Pic}(C_\varepsilon)\rightarrow \mathrm{Pic}(C))=T_0\mathrm{Pic}_C$

For smooth projective curve $X$ over $\overline{k}$ with $\ell$ an invertible prime, $H^1(X,\mu_{\ell^n})=\mathrm{Pic}(X)[\ell^n]$.

Proof.

Start with Kummer sequence

$$1 \rightarrow \mu_{\ell^n} \rightarrow \mathbf{G}_m \xrightarrow{\ell^n} \mathbf{G}_m \rightarrow 1$$

Deriving cohomology

$$H^0\left(X, \mathbf{G}_m\right) \xrightarrow{\ell^n} H^0\left(X, \mathbf{G}_m\right) \rightarrow H^1\left(X, \mu_{\ell^n}\right) \rightarrow H^1\left(X, \mathbf{G}_m\right) \xrightarrow{\ell^n} H^1\left(X, \mathbf{G}_m\right)$$

Thus $H^1\left(X, \mu_{\ell^n}\right) \cong H^1(X,\mathbf{G}_m)[\ell^n]\cong \operatorname{Pic}(X)\left[\ell^n\right]$.

For a smooth projective curve $C$ of genus $g$ and its Jacobian $J=\mathrm{Jac}(C)$, we have an isomorphism of $\mathrm{Gal}_k$-modules

$$i^*:H^1(C_{\overline{k}},\mathbb Q_\ell)\cong H^1(J_{\overline{k}}, \mathbb Q_\ell)$$

induced by the Abel-Jacobi map $i:C\to J$ defined as $P\mapsto [P-P_0]$ with $P_0$ a rational point.

Proof.

We have $H^1\left(C, \mu_{\ell^n}\right) \cong \mathrm{Pic}(C)\left[\ell^n\right] \cong\mathrm{Pic}^0(C)\left[\ell^n\right] \cong J\left[\ell^n\right]$. Similarly, $H^1(J,\mu_{\ell^n})=\mathrm{Pic}^0(J)[\ell^n]=J^\vee [\ell^n]$ via the principal polarization.

L-poly, Jacobian,

Semilinear Algebra and Hasse–Witt Matrix

Let $\varepsilon$ be an automorphism of the field $K$, and write $a^{\varepsilon}:=\varepsilon(a)$ for all $a \in K$. A map $T: V \to W$ between two $K$-vector spaces is called $\varepsilon$-semilinear if it satisfies the following properties for all $v, w \in V$ and $a \in K$: $T(v + w) = T(v) + T(w)$ and $T(av) = a^{\varepsilon} T(v)$. Let $E=(e_1, \ldots, e_n)$ and $F=(f_1,\dots,f_n)$ be an ordered basis for $V$, we represent a $\varepsilon$-semilinear map $T$ by a matrix $[T]_{E}^{F}$. Unlike in linear algebra, we have $[Tv]_{F} = [T]_{E}^{F} [v]_{E}^{\varepsilon}$, where $[v]_{E}^{\varepsilon}$ is obtained by applying $\varepsilon$ to each entry of $[v]_{E}$, and change of basis is: $[T]_{F}^{F} = [\mathrm{id}]_{E}^{F} [T]_E^E([\mathrm{id}]_{E}^{F})^{\varepsilon}$. Iteration is given by $[T^n]_{E}^{E} = [T]_E^E ([T]_E^E)^{\varepsilon} \cdots ([T]_E^E)^{\varepsilon^{n-1}}$.

Let $V^*$ be the dual of $V$ and $E^*=(e^1, \ldots, e^n)$ be the corresponding dual basis. If $T: V \to W$ is a $\varepsilon$-semilinear map, then its adjoint $T^*: W^* \to V^*$ is a $\varepsilon^{-1}$-semilinear map and is characterized by the property that $\langle Tv, w \rangle = \langle v, T^* w \rangle^{\varepsilon^{-1}}$ for all $v \in V$ and $w \in W^*$. In terms of matrices, we have $[T^*]_{F^*}^{E^*} = ([T]_E^F)^{t,\varepsilon^{-1}}$, where $t$ denotes the transpose.

Let $k$ be a perfect field of characteristic $p > 0$, and let $X$ be a smooth projective curve over $k$. Let $F: X \to X$ be the absolute Frobenius morphism, which raises functions to their $p$-th power. Let $\sigma:k\rightarrow k$ be the Frobenius automorphism of $k$ with inverse $\tau$. The absolute Frobenius $F:X \to X$ is defined to be the identity on the underlying topological space of $X$ and sends $f\mapsto f^p$ on sections. Recall that the coherent cohomology $H^i(X,\mathcal O_X)$ is a $k$-vector space via the following process: the structure map $X\rightarrow \mathrm{Spec}(k)$ induces $k=H^0(\mathrm{Spec}(k),\mathcal O_{\mathrm{Spec}(k)})\rightarrow H^0(X,\mathcal O_X)$, and for each $f\in H^0(X,\mathcal O_X)$, the multiplication map $m_f:\mathcal O_X \to \mathcal O_X$ by $f$ induces $m_f^i: H^i(X,\mathcal O_X) \to H^i(X,\mathcal O_X)$, and $f\cdot \alpha := m_f^i(\alpha)$ for $\alpha \in H^i(X,\mathcal O_X)$ makes $H^i(X,\mathcal O_X)$ a $H^0(X,\mathcal O_X)$-module, and hence by composition also a $k$-vector space. The absolute Frobenius $F$ induces a $\sigma$-semilinear map $F^*: H^1(X,\mathcal O_X) \to H^1(X,\mathcal O_X)$.

Let $B$ be a basis of $H^1(X,\mathcal O_X)$, the Hasse–Witt matrix of $X$ with respect to $B$ is defined to be the matrix representation of the $\sigma$-semilinear map $F^*$ with respect to $B$, i.e. $[F^*]_B^B$.

Let $\Omega^1_X$ be the sheaf of Kähler differentials on $X$. We introduce another matrix which is closely related to the Hasse-Witt matrix called the Cartier-Manin matrix, to which end we need to define the Cartier operator.

The Cartier operator $C:H^0(X,\Omega^1_X) \to H^0(X,\Omega^1_X)$ is defined as the semilinear adjoint of the induced map by the absolute Frobenius $F^*: H^1(X,\mathcal O_X) \to H^1(X,\mathcal O_X)$ with respect to the Serre duality pairing, i.e. $\langle F^* \alpha, \omega \rangle = \langle \alpha, C\omega \rangle^{\tau}$ for all $\alpha \in H^1(X,\mathcal O_X)$ and $\omega \in H^0(X,\Omega^1_X)$.

Concretely, we can derive a formula for the Cartier operator as follows.