Theta Characteristics and Quadrics in Characteristic Two

Last time we defined theta characteristics as square roots of the canonical bundle.  Today, we’re going to analyze the notion a bit, and relate them to quadrics in characteristic two.

We start by denoting the set of theta characteristics on a curve $C$ by $\mathrm{TChar}(C)$.  The first nice property of this set is that it’s a torsor over the points of order two on $J(C)$.  We have a natural action of $J(C)[2]$ on $TChar(C)$ by $(\mu,L)\mapsto \mu\otimes L$.  This will still be a theta characteristic, because after squaring, $\mu^2\otimes L^2\cong L^2\cong \omega_C$.   This action is free, and it is transitive, because if $L,L'$ are theta characteristics, then $L'\otimes L^{-1}$ squares to the trivial line bundle, and so they differ by some point of order two.

This argument tells us that there are $2^{2g}$ theta characteristics of $C$.  We then split them into two types, even and odd, distinguished by the parity of $h^0(C,L)$ with $L$ a theta characteristic.  But how many of each are there? We’re going to spend the rest of this post computing these numbers.

A quadratic form on a vector space is a function $q:V\to k$ such that for all $v\in V, a\in k$, we have $q(av)=a^2q(v)$, and such that the map $b:V\times V\to k$ given by $q(v+w)-q(v)-q(w)$ is bilinear.  Then, $b(v,v)=2q(v)$.  As we’re going to be looking in characteristic two, this implies that $b(v,v)=0$, and so $b$ is a symplectic form.  Now, we fix a basis $e_1,\ldots,e_n$ of $V$, and define $A=(a_{ij})=(b(e_i,e_j))$.  Then we can write $q(\sum x_ie_i)=\sum x_i^2q(e_i)+\sum x_ix_ja_{ij}$.  We restrict to looking only at nondegenerate $q$, where $\mathrm{rank}(A)=\dim V$.

Now, as $b$ is symplectic and $q$ is nondegenerate, we must have $n=2g$, and we can choose a basis such that $A=\left(\begin{array}{cc} 0 & I_k \\ I_k & 0 \end{array}\right)$, so we can write $q(\sum x_ie_i)=\sum x_i^2 q(e_i)+\sum x_i x_{i+g}$.  We only care about the field of two elements, in fact, so every element is its own square root, so $q(\sum x_ie_i)=(\sum x_iq(e_i))^2+\sum x_ix_{i+g}$.

We can define a chosen quadratic form by $q_0(\sum x_ie_i)=\sum x_ix_{i+g}$.  Then, given a form and a vector, we can write $(q+\eta)(v)=q(v+\eta)+q(\eta)$, so the quadrics are a torsor over the vector space itself.  We define the Arf invariant by $\mathrm{Arf}(q)=\sum q(e_i)q(e_{i+g})$.  It turns out that, as any $q$ can be written uniquely as $q=q_0+\eta_q$, the Arf invariant is $q(\eta_q)=q_0(\eta_q)$, and this value is called the parity of the form, that is, even or odd.

So now, we can compute the number of even and the number of odd quadratic forms.  It’s just $|q_0^{-1}(0)|$ and $|q_0^{-1}(1)|$.   Computing the number is actually a straightforward induction argument.  If $g=1$, then the cardinalities are $3$ and $1$.  In general, they are $2^{g-1}(2^g+1)$ and $2^{g-1}(2^g-1)$.

So, how does this connect to theta characteristics? Fix a theta characteristic $L$.  Then it defines a quadratic form on the $\mathbb{F}_2$-vector space $J(C)[2]$, defined by $q(\mu)=h^0(\mu\otimes L)+h^0(L)\mod 2$.  These will be even or odd based on whether $L$ is, and so we’ve computed the number of even and odd theta characteristics!