I've seen the following sentence come up in a few papers:



> 
> Consider the modular curve $Y\_1(N)$ and let $E$ be the universal elliptic curve over $Y\_1(N)$.
> 
> 
> 


This comes up in Deligne's construction of Galois representations for modular forms of weight $k>2$. I'm not sure what "universal elliptic curve" means, and after some googling, I can't seem to find a complete definition / construction of it. So I wanted to ask:


1. Does anyone have a reference which explains the full definition of "universal elliptic curve" (over a modular curve)? Preferably, I'd like a reference which explains the overall philosophy / motivation behind universal elliptic curves, as well as the rigorous definitions.
2. How does one write down an explicit Weierstrass equation for the universal elliptic curve over $Y\_1(N)$? For example, if we set say $N=11$, then what is the Weierstrass equation for the universal elliptic curve $E$ over $Y\_1(11)$?



Reference for universal elliptic curves
For any $n\geq 1$, one can define a functor $\mathcal{M}\_1(n)\colon \mathrm{Schemes}/\mathbb{Z}[\frac1n] \to \mathrm{Groupoids}$, sending a scheme to the groupoid of elliptic curves over it with a chosen point of exact order $n$; the morphisms in the groupoid are the isomorphisms of elliptic curves that respect the chosen point. If $n\geq 4$, the groupoid is equivalent to a set (i.e. there can be no non-trivial automorphism of an elliptic curve that preserves a point of order $n\geq 4$) and the functor is actually represented by a scheme. This scheme might be called $Y\_1(n)$. If you define it like this, the universal elliptic curve over $Y\_1(n)$ is simply classified by the identity. (For $n<4$, the scheme $Y\_1(n)$ is the coarse moduli space of the stack $\mathcal{M}\_1(n)$ and the story becomes more complicated.)


But maybe $Y\_1(n)$ has another definition for you, e.g. as the quotient of the upper half plane $\mathbb{H}$ by $\Gamma\_1(n)$. It is a non-trivial theorem to identify this with the analytification of the $Y\_1(n)$ as constructed above. But you can construct the universal elliptic curve in this case also easily in a direct way: Just take the quotient of $\mathbb{H} \times \mathbb{C}$ by the suitable semidirect product of $\Gamma\_1(n)$ and $\mathbb{Z}^2$.


Probably, Katz's [$p$-adic properties of modular schemes and modular forms](https://web.math.princeton.edu/%7Enmk/old/padicpropMFMS.pdf) is not the worst place to read about this. From the purely complex point of view you also have Hain's [Lectures on Moduli Spaces of Elliptic Curves](https://arxiv.org/abs/0812.1803).


Regarding explicit Weierstraß equations: in general, this is bound to be difficult as even explicit equations for the $Y\_1(n)$ are hard to get. An older article which deals in particular with such equations is *Torsion groups of elliptic curves with integral j-invariant over quadratic fields* by Hans H. Müller, Harald Ströher and Horst G. Zimmer. Regardless of the precise equations for $Y\_1(n)$ ones knows however that the universal elliptic curve over it always has a Weierstraß equation of the form
$$y^2 + a\_1xy + a\_3y = x^3 +a\_2x^2,$$
for holomorphic modular forms $a\_1, a\_2$ and $a\_3$ of level $n$ over $\mathbb{Z}[\frac1n]$. This might have been known before, but we give a proof in an article with V. Ozornova: [Rings of modular forms and a splitting for $TMF\_0(7)$](https://arxiv.org/abs/1812.04425). There we also work out the case $n=7$ in detail.


