Cassels [@cassels1], and by Schaefer [@schaefer1], and Poonen-Schaefer [@schaefer2] in more generality. Throughout the rest of this section we will focus on computing the dimension of the 2-Selmer group of the jacobian $J$ of a hyperelliptic curve $C$, given by an affine equation of the form $$C:y^2=f(x),$$ where $f\in{\mathbb Z}[x]$ is square-free and $\deg(f)$ is odd (Stoll also treats the case when $\deg(f)$ is even, but we do not need it for our purposes). In this case, the curve $C$ is of genus $g=(\deg(f) - 1)/2$ with a single point at infinity in the projective closure. Before we can compute the dimension of the 2-Selmer group, we must define a few objects of interest and examine some of their properties. We will follow the notation laid out in [@stoll].

Let ${{\rm Sel}}^{(2)}({\mathbb Q},J)$ be the 2-Selmer group of $J$ over ${\mathbb Q}$, and let $\Sha({\mathbb Q},J)[2]$ be the $2$-torsion of the Tate-Shafarevich group of $J$ (as defined, for instance, in Section 1 of [@stoll]). Selmer and Sha fit in the following fundamental short exact sequence: $$\xymatrix{
0\ar[r]& J({\mathbb Q})/2J({\mathbb Q})\ar[r]& {{\rm Sel}}^{(2)}({\mathbb Q},J)\ar[r]& \Sha({\mathbb Q},J)[2]\ar[r]& 0.
}$$ With this sequence in hand we get a relationship between for the rank 