ag.algebraic geometry - Cohomology of quaternions on an abelian variety

For the description of the quaternion algebra associated to a pair of torsion line bundles, try the following. Take line bundles ${cal L}_i$ on $E_i$ equipped with isomorphisms ${cal L}_i^{otimes 2} to {cal O}$, and pull these back to $A$. Define

$$D = {cal O} oplus {cal L}_1 oplus {cal L}_2 oplus {cal L}_1 otimes {cal L}_2$$

with multiplication induced by the maps ${cal L}_i^{otimes 2} to {cal O}$, ${cal O}$ being the unit, and the elements of ${cal L}_1$ and ${cal L}_2$ anticommuting.

ADDED: I wish I had a more conceptual explanation for why this represents the cup-product in the Brauer group, but here is a cocycle description along the lines of what Oren suggested.

Suppose $X$ is given with 2-torsion line bundles $cal L$ and $cal M$. Choose cocycles representing these, in the form of a cover (either open in the analytic case, or etale in the algebraic case) $U_alpha$ of $X$ together with sections $s_alpha$ of $cal L$ and $t_alpha$ of $cal M$ on $U_alpha$ such that $s_beta / s_alpha = u_{alpha beta} in {pm 1}$ and similarly $t_beta / t_alpha = v_{alpha beta}$; these latter two are the representing cocycles.

Then D has basis ${1,s_alpha, t_alpha, s_alpha t_alpha}$ on $U_alpha$, where $s_alpha^2 = t_alpha^2 = 1$, and you can explicitly make this isomorphic to a matrix algebra. The change-of-basis sends $s_alpha$ to $s_beta = u_{alpha beta} s_alpha$ and similarly for $t$. This can be achieved by conjugation by the element $t_alpha^{(1 - u_{alpha beta})/2} s_alpha^{(1 - v_{alpha beta})/2} = g_{alphabeta} in D cong M_2(mathbb{C})$. These change-of-basis matrices reduce to a cocycle in $PGL_2(mathbb{C})$ representing the algebra, and so the image in the Brauer group is represented by the coboundary $(delta g)_{alpha beta gamma} in {pm 1}$.

EDIT: fixed up following description of the coboundary so that it correctly described where the cup product lands.

Explicit computation finds $(delta g)_{alpha beta gamma}$ is equivalent to the cocycle $v_{alpha beta} otimes u_{beta gamma}$ with coefficients in ${pm 1} otimes {pm 1} cong {pm 1}$ (I may have mixed the indices, if I did please let me know and I'll correct it), which is precisely the formula for the cup product of the cocycles $u$ and $v$.

So based on your description of the cup product inducing an isomorphism between 2-torsion in the Brauer group and cup products of 2-torsion elements in the Picard group, this genuinely should provide you with the bundles you're looking for.