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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 $Y_1(11)$?

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  • $\begingroup$ A nice reference is "Modular Forms and Modular Curves" by Diamond and Im, available here: people.math.wisc.edu/~boston/…. The second paragraph of page 32 explains universal elliptic curves very nicely. $\endgroup$ Jul 8 at 19:16

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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 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.

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 $TMF_0(7)$. There we also work out the case $n=7$ in detail.

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  • $\begingroup$ Regarding the existence of Weierstraß equations, I asked once a related question about the triviality of the canonical bundle of the universal elliptic curve, maybe you can answer it. $\endgroup$ Jun 21 at 11:57
  • $\begingroup$ wonderful! thanks for the description and the references, they were very helpful. $\endgroup$ Jun 23 at 13:24
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To complete Lennart Meier's nice answer, Baaziz has computed explicit equations for the universal elliptic curve over $Y_1(N)$ up to $N=51$. The method used and the data up to $N=20$ can be found in Baaziz's article.

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