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In Ogg's paper "rational points on certain elliptic modular curves", the author says, using Ogg's notation for cusps, that for fiexed $d$, if $(y, N) = d$, then for any $x$ satisfying $(x, y, N) = 1$, the cusps of the form $$\left( \begin{array}{c} x \\ y \end{array} \right) $$ are conjugate, and hence for $d = 1, 2$, these are rational.

But I don't know the moduli interpretation for cusps in $X_1(N)$ for $N$ composite.

I know that the cusps $$\left( \begin{array}{c} 0 \\ y \end{array} \right) $$ are corresponding to the Neron $N$-gon with a $\Gamma_1(N)$-structure $(1, a) \in \mu_N \times \mathbb{Z}/N$, and the cusps $$\left( \begin{array}{c} x \\ 0 \end{array} \right) $$ are $1$-gon.

How about other cusps?

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    $\begingroup$ This is explained in Diamond-Im, Modular forms and modular curves, section 9.3. You should get the Galois action from that. $\endgroup$
    – François Brunault
    Commented Feb 26, 2021 at 13:21
  • $\begingroup$ @FrançoisBrunault Thank you. I checked that, but couldn't find it. In 9.3.4, the authors say that the singular locus of the Tate curve with $1$-gon, resp., $p$-gon gives $\infty$, resp., $0$ without proof. I want to know its proof, and the case of composite levels. $\endgroup$
    – k.j.
    Commented Feb 26, 2021 at 17:18
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    $\begingroup$ This is an important point. For $Y_1(N)$ here is the idea: $\mathbb{C}/(\mathbb{Z}+\tau \mathbb{Z})$ has algebraic Weierstrass equation $y^2 = x^3 - g_4(\tau) x - g_6(\tau)$ where $g_4$ and $g_6$ are $\textit{holomorphic}$ functions of $\tau$. For details see the notes by Loeffler warwick.ac.uk/fac/sci/maths/people/staff/fbouyer/…, Prop 3.13. Similar for $X_1(N)$, using the Tate curve $y^2+xy = x^3 + a_4(q) x + a_6(q)$. When $q \to 0$ this converges to a $1$-gon (see Diamond-Im, 9.2). For other cusps, use level structures on the Tate curve over $\mathbb{C}((q^{1/N}))$. $\endgroup$
    – François Brunault
    Commented Feb 28, 2021 at 8:19
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    $\begingroup$ The Tate curve provides a description of the complex points of the universal generalised curve $E_1(N) \to X_1(N)$ near the cusps (the cusps correspond to level structures on the Tate curve, see Loeffler, Prop 4.6). The map $(\tau,z) \mapsto (X(u,q),Y(u,q))$ (notations in Prop 4.5) shows that $E_1(N)(\mathbb{C})$ is a Néron polygon above the cusps, and that $X_1(N)(\mathbb{C})$ is what you think. Unfortunately I don't have the time to give a lecture on this so I suggest you read the references I gave, and work them out. $\endgroup$
    – François Brunault
    Commented Mar 1, 2021 at 10:19
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    $\begingroup$ Maybe also this introduction: math.leidenuniv.nl/~pbruin/moduli.pdf $\endgroup$
    – François Brunault
    Commented Mar 1, 2021 at 10:21

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