Explicit equations for the universal vector extension of an elliptic curve

Question

The universal vector extension $E$ of an abelian variety $A$ is an algebraic group, an extension of $A$ by a vector group $0 \to V \to E \to A \to 0$, such that any other extension of $A$ by a vector group $0 \to V' \to E' \to A \to 0$ arises as a pushout $E' \cong V' \times^V E$ for some morphism from $V \to V'$. See https://www.iazd.uni-hannover.de/fileadmin/iazd/Gao/Oberwolfach2016.pdf, section 1.1. I am told that $E$ can also be thought of as a moduli space of vector bundles on $A$ equipped with an integrable connection, and it has the structure of a quasi-projective variety. I am interested in finding explicit equations for $E$ in the case where $A$ is an elliptic curve over $\mathbb{C}$ (my ultimate interest is when $A$ is an elliptic curve over a $p$-adic field, but I can forgo that for now). It would be especially nice if the map from the moduli-theoretic interpretation of $E$ to the explicit variety can be made explicit. Is this worked out anywhere in the literature, or is it straightforward to work it out for oneself from the available literature? Even the special case of $A$ a single specific elliptic curve would be helpful. Thank you.

This looks related to this previous unanswered question: Universal vectorial bi-extension as a scheme

Answer 1

(Not a complete answer but too long for a comment)

My guess is that such a description is not known, probably because there isn't an easy one. There is a very nice complex analytic description as $\mathbb{C}^2$ modulo a rank two lattice of the form (periods,quasiperiods). It shouldn't be too hard to convert this description into a $p$-adic analytic one for the $p$-adic Tate curve, which might be useful for you.

In characteristic $p>0$, I gave a description of the universal vectorial extension of an ordinary elliptic curve as a quotient of $A^{(p)} \times \mathbb{G}_a$ (where $A^{(p)}$ is the image of $A$ under Frobenius) by an action of a group of order $p$ and, from this, it shouldn't be too hard to write down explicit equations, but they will very much depend on $p$.

Both cases (which are in a way parallel) come with the moduli-theoretic interpretation you require. In my paper, I also describe the situation over $\mathbb{C}$ (with references) and make the parallels explicit.

Reference:

An analogue of the Weierstrass zeta-function in characteristic p, Acta Arithmetica, LXXIX (1997) 1-6.