Let $(E,O)$ be an elliptic curve over $\mathrm{Spec}(\mathbb{C})$. Then a 1-dimenstional represenation of $\pi_1(E^{\mathrm{an}}) = \mathbb{Z} \times \mathbb{Z}$ over $\mathbb{C}$ is just a pair of elements $(a, b)$ of $\mathbb{C}^\times$. Under a standard equivalence of categories, the representation corresponding to a pair $(a, b)$ corresponds to an analytic connection on $E^{\mathrm{an}}$ which we'll denote by $L^{\mathrm{an}}_{(a,b)}$ (see, e.g., theorem 4.2.4 in the book "D-modules, perverse sheaves and representation theory" in the book by Hotta et al). One description of $L^{\mathrm{an}}_{(a,b)}$ is given in Mariano's answer below.
Now we have a equivalence of categories between analytic connections on $E^{\mathrm{an}}$ and algebraic connections on $E$ (see, e.g., corollary 5.3.9 in Hotta et al.).
I'd like to know an explicit description of the algebraic connection $L_{(a,b)}$ on $E$ corresponding to the analytic connection $L^{\mathrm{an}}_{(a,b)}$ on $E^{\mathrm{an}}$ under this equivalence.
The first step would be to determine the underlying invertible $\mathscr{O}_E$-module, which should be of degree 0 (see, e.g., Atiyah's "Complex Analytic Connections in Fiber Bundles," proposition 19). But $\mathrm{Pic}^0(E) = E(\mathbb{C})$ since $E$ is an elliptic curve. In other words, we get a map $$(\mathbb{C}^\times)^2 \to E(\mathbb{C}),$$ taking $(a,b) \in (\mathbb{C}^\times)^2$ to the closed point $P_{(a,b)} \in E(\mathbb{C})$ such that $\mathscr{O}_E(P_{(a,b)}-O)$ is the line bundle underlying the connection $L_{(a,b)}$. What is $P_{(a,b)}$ in terms of $(a, b)$? Given $P \in E(\mathbb{C})$, what are the $(a,b) \in (\mathbb{C}^\times)^2$ such that $P = P_{(a,b)}$?
How can we describe the connection on $L_{(a,b)}$ in terms of $(a, b)$? We know that $\Omega^1_E$ has a global nowhere-vanishing section $\omega$. Let $\partial$ be the derivation of $\mathscr{O}_E$ dual to $\omega$. We know that $L_{(a,b)}$ is trivial on $E \setminus \{O, P_{(a,b)}\}$, so the connection on this open is given by a differential equation of the form $\partial - f$ for some $f \in \Gamma(U, \mathscr{O}_E)$. Can we explicitly write down such a $f$?
$\def\ZZ{\mathbb{Z}}\def\CC{\mathbb{C}}$Let $X=\CC\times\CC$, fix $\tau$ in the open upper halfspace, and let $a$ and $b$ be in $\CC^\times$. There is on $X$ an action of $\ZZ\times\ZZ$ such that we have $$(x,y)\cdot(z,v)=(z+x+\tau y,a^xb^yv)$$ whenever $(x,y)\in\ZZ\times\ZZ$ and $(z,v)\in E$. The first projection $\pi:X\to\CC$ is equivariant for the usual action of $\ZZ\times\ZZ$ on $\CC$, so by passing to the quotient we end up with a map $\bar\pi:V=X/(\ZZ\times\ZZ)\to E$, with $E$ an elliptic curve, and this is easily seen to be a line bundle on $E$. There is a connection on $V$ whose horizonal sections on the open subsets of $E$ well covered by the projec tion $\CC\to E$ are precisely the constant sections of $\pi$. This means that the Christoffel symbols of the connection in charts induced from the covering $\CC\to E$ are zero.