Given a natural number $g$, there is a natural representation $R_{g}:B_{2g+1}\to\text{Sp}(2g,\mathbb{Z})$ of the braid group on $2g+1$ threads within the $2g\times 2g$ symplectic matrices, known as the symplectic representation (see, for instance, here). Explicitly, we can consider a $(2g+2)$-punctured sphere and consider its double cover with branchings at the punctures, which is a genus $g$ Riemann surface $\Sigma_g$. The action of the braid group on the punctured sphere (keeping one point fixed) induces an action on the homology group $H_1(\Sigma_g,\mathbb{Z})$ of the double cover which preserves the intersection pairing $\langle\cdot,\cdot\rangle$ and thus is a representation within $\text{Sp}(2g,\mathbb{Z})\cong\text{Aut}(H_1(\Sigma_g,\mathbb{Z}),\langle\cdot,\cdot\rangle)$.
My question is: what is known about the image $R_g(B_{2g+1})\subset\text{Sp}(2g,\mathbb{Z})$ of the braid group within the symplectic matrices?
For $g=1$ it is possible to be very explicit and we find that, if $\sigma_1$ and $\sigma_2$ are the generators of $B_3$, then $$R_1(\sigma_1)=\begin{pmatrix}1 & 1\\0 & 1\end{pmatrix}\,,\quad R_1(\sigma_2)=\begin{pmatrix}1 & 0\\ -1 & 1\end{pmatrix}\,,$$ which are known to generate the full modular group $\text{Sp}(2,\mathbb{Z})\cong\text{SL}(2,\mathbb{Z})$ (in fact, $B_3$ is known to be the universal central extension of $\text{PSL}(2,\mathbb{Z})$).
What is known at $g>1$?
So the representation factors through the mapping class group of that genus with each half-twist $\sigma_i$ representing in the MCG as a Dehn twist and in the symplectic group as a transvection. In the $g=2$ case these 'horizontal' Dehn twists in $MCG_2$ generate the whole MCG, and so their image in the Symplectic group is all of the Symplectic group.
For $g>2$ these horizontal Dehn twists do not generate all of $MCG_g$. In fact the subgroup of the Symplectic group generated by the corresponding transvections does not map onto $Sp(2g,\mathbb Z_2)$ when you reduce coefficients mod $2$.