I’m going to describe some ongoing work with a friend of mine, John Halliday. We’re working under David Zureick-Brown.

The project concerns the moduli of abelian varieties; specifically, the goal is to provide formulae for the Frobenius and Verschiebung on, and consequently compute a certain invariant of, the Jacobian of a hyperelliptic curve in odd characteristic. The motivation for this project comes from the Hasse invariant.

There are multiple equivalent ways to state that an elliptic curve $E$ defined by $y^2 = f(x)$ over a field $k$ (of characteristic $p$) is ordinary, some of which are as follows.

• The formal group associated to $E$ is of height $1$, i.e., is isogenous to $\mathbf{G}_m$.
• We have $E[p^k]\simeq \mathbf{Z}/p^k$ for all $k$.
• The trace of Frobenius on $E$ is nonzero.
• The Newton polygon of $E$ is ‘‘as low as possible’’, i.e., has a line of slope $-1$ from $(0,1)$ to $(1,0)$, and has a line of slope $0$ from $(1,0)$ to $(2,0)$.
• Let $H_p$ denote the coefficient of $x^{p-1}$ in $f(x)^{(p-1)/2}$. Then $H_p\neq 0$.

The last invariant is called the Hasse invariant (one can show that the Hasse invariant is congruent, modulo $p$, to the trace of Frobenius), generalizations of which will be the focus of this post.

Supersingular elliptic curves admit a similar characterization:

• The formal group associated to $E$ is of height $2$.
• We have $E[p^k]\simeq 0$ for all $k$.
• The trace of Frobenius on $E$ is zero.
• The Newton polygon is a line of slope $1/2$.
• The Hasse invariant is zero.

The Hasse invariant is easy to compute, but the way it’s defined doesn’t give intuition as to why this is the correct definition. To explain the definition, we need to set some notation. Let $F$ denote the Frobenius on $E$; this is a morphism of schemes $E\to E^{(p)}$. The dual of this is the Verschiebung, which is a morphism $V:E^{(p)}\to E$.

The Verschiebung induces a morphism on (co)tangent spaces, and in particular, gives a morphism $V:\Omega^1_{K^p} \to \Omega_K$, where $K = K(E)$ and $K^p$ is $K(E^{(p)})$. (Note that the Frobenius also gives a morphism $\Omega^1_K\to \Omega^1_{K^p}$.) We can compute this map $V$ explicitly by fixing a nice basis for the differential forms of degree $1$ and of the first kind (these are exactly the global sections of $\Omega^1_K$). This is just a one-dimensional $k$-vector space, and choose the basis consisting of the vector $dx/y$, which we’ll denote by $\omega$.

To determine $V$, it suffices to determine $V(\omega)$. Explicitly, we find that we can write any element of $\Omega^1_{K^p}$ in the form $d\phi+\eta^p x^{p-1} dx$, where $\phi,\eta\in K$ and $\eta^p\in K^p$, and that $V$ sends this to $\eta \ dx$ (exactly as one would expect the Verschiebung to behave if it’s to be a dual to the Frobenius!). Computing $V(\omega)$ will be a minute’s walk away if we can write $\omega$ in this form.

This is easy: define numbers $c_i$ by the expansion $f(x)^{(p-1)/2} = \sum^{3(p-1)/2}_{j=0}c_j x^j dx$. Then

One can then compute (we’ll do this in more generality below) that $V(\omega) = c^{1/p}_{p} dx/y$. Similarly, one finds that $F(\omega) = c_p d^px^p/y^p$, where $d^p:K^p\to \Omega^1_{K^p}$ is the universal derivation.

In particular, the nonvanishing of the Hasse invariant is equivalent to the tangent map of the Verschiebung $V:E^{(p)} \to E$ being nonzero. This in particular means that the multiplication-by-$p$ map (which is the Verschiebung composed with the Frobenius) will be some sort of covering of degree $p$ — and one can show that this is equivalent to $E[p]$ having $p$ connected components.

The method of generalization to hyperelliptic curves is an exact analogue of this discussion. Let $C:y^2 = f(x)$ now be a hyperelliptic curve of genus $g$, so that the degree of $f$ is $2g+1$. Then – exactly as above – we can consider the effect of the Verschiebung and the Frobenius on the sheaf of Kahler differentials. This gives what are known as the ‘‘modified Cartier’’ and the ‘‘Cartier’’ operators, respectively. We may pick the basis (as a $k$-vector space) for the global sections of $\Omega^1_{C}$ given by $\omega_i := x^{i-1} dx/y$ for $1\leq i\leq g$. We can rewrite

and then attempt to compute the effect of applying $V$ and $F$.

Again, we find that we can write any element of $\Omega^1_{K^p}$ in the form $d\phi+\eta^p x^{p-1} dx$, where $\phi,\eta\in K$ and $\eta^p\in K^p$, and that $V$ sends this to $\eta \ dx$.

Lemma: The action of $F$ and $V$ on $H^0(C,\Omega^1_E)$ is determined by the following equations.

Proof. It’s clear that $F(\omega_i) = 0$, so we’ll just compute $V(\omega_i)$. This computation will be immediate if we can write $\omega_i$ in the form $d\phi+\eta^p x^{p-1} dx$, which we can do: since $\omega_i = y^{-p} \sum^{(p-1)/2\cdot (2g+1)}_{j=0} c_j x^{j+i-1} dx$, we get

It follows that

as desired.

Define a matrix $A$ via $A_{i,j} = c_{ip-j}$; then, if $A^{(1/p)}$ denotes the matrix such that $A^{(1/p)}_{i,j} = \left(A_{i,j}\right)^{1/p}$, the Verschiebung $V$ acts via $A^{(1/p)}$. This matrix is the Hasse-Witt matrix. It’s clear that it’s unique up to transformations of the form $A\mapsto S^{(p)} A S^{-1}$, where $S^{(p)}_{i,j} = (S_{i,j})^p$.

An important application of this construction is the following theorem.

Theorem (Yui): Let $C$ be a hyperelliptic curve of genus $g$ over $k$. If the Hasse-Witt matrix of $C$ is $0$ in $k$, then the Jacobian $J(C)$ is supersingular and is isogenous to $g$ copies of a supersingular curve (over some finite extension of $k$).

Notice that $F$ and $V$ both take exact forms to zero. In fact, the Hasse-Witt matrix $A$ is exactly the matrix (in that basis written down above) for the action of the Frobenius on $H^1(C,\mathcal{O}_C)$. Equivalently, by Serre duality, it is the matrix of the Verschiebung on the zeroth cohomology of the de Rham complex.

There is a larger vector space which contains more information than just $H^1(C,\mathcal{O}_C)$ and $H^0(C,\Omega_C)$. This is the (first) de Rham cohomology of $C$. Recall that the Hodge–de-Rham spectral sequence runs

The resulting filtration on $H^\ast_{dR}(X)$ is called the Hodge filtration. There’s a hard theorem, which is established with analytic techniques:

Theorem: The Hodge–de-Rham spectral sequence collapses at the $E_1$-page if $X$ is smooth and proper.

This is fabulous, because now the associated graded of the Hodge filtration is exactly $\mathrm{gr}^p H^i_{dR}(X) = H^{i-p}(X,\Omega^p_{X/K})$. Let $h^{p,q} = \dim_K H^q(X,\Omega^p_{X/K})$. Serre duality gives us $h^{p,q} = h^{n-p,n-q}$, where $n$ is the dimension of $X$. One also has “conjugation symmetry”, which says that $h^{p,q} = h^{q,p}$.

We can, in particular, consider the Hodge filtration for $H^1_{dR}(X)$. This is rather easy; given the description above, we have a short exact sequence

The two associated graded are of the same dimension.

As above, suppose that $X=C$ was a smooth geometrically connected projective hyperelliptic curve over $k$ of genus $g$. It follows from GAGA that $H^1_{dR}(X)$ is a $2g$-dimensional $k$-vector space. Since $C$ lives over $k$, we still have an action of $F$ and $V$ on $H^1_{dR}(C)$, and so we might ask if there’s any additional information that could be gained if we were to study this action on this larger cohomology group. One way to concretely phrase this desire is through the following question.

Question: Can we write down the action of the Frobenius (and the Verschiebung) on $H^1_{dR}(C)$? This is an approximation to a harder question of computing the action of $F$ and $V$ on the crystalline cohomology of $C$ (which reduces mod $p$ to de Rham cohomology); so: equivalently, can we compute the Dieudonne module of $C$ modulo $p$?

The goal of our project is to answer this question. As a consequence of performing this computation, we’ll also be able to describe the “Ekedahl-Oort type” (or at least the canonical type) of Jacobians of hyperelliptic curves. I’ll describe this computation, and the final type, in the next post.