Let $k$ be a number field , $f \in k[x]$ of degree $2g$ with distinct roots, and $X$ the complex plane with the roots of $f$ removed. Then define the abelian scheme $J \rightarrow X$ where for each $t \in X$, $J_{t}$ is the jacobian of the hyperelliptic curve $y^{2} = f(x)(x - t)$. Then there is a monodromy representation of the topological fundamental group $\rho: \pi_{1}(X) \rightarrow \mathrm{Sp}_{2g}(\mathbb{Z})$ which preserves the Riemann form on the homology elements of the fiber over the basepoint.

I know that the image of this representation is the subset of $\mathrm{Sp}(\mathbb{Z})$ whose image modulo 2 is the identity but don't know of any "elementary" proof of it. In particular, I've seen it claimed that for genus $g = 1$, the image of the representation is simply $\Gamma(2) \cap \mathrm{SL}_{2}(\mathbb{Z})$. Can anyone tell me how to prove it for this elliptic curve case, in an intuitive way? I'm looking for something similar to http://rigtriv.wordpress.com/2010/02/25/monodromy-representations/, which I can't quite follow.

I would very much appreciate any kind of argument, but especially a relatively intuitive, visual one.