Timeline for Moduli interpretation and Ogg's notation for the cusps on modular curves
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Mar 1, 2021 at 10:31 | comment | added | k.j. | @FrançoisBrunault I’ll try it. Much appriciated for your kindness! | |
| Mar 1, 2021 at 10:21 | comment | added | François Brunault | Maybe also this introduction: math.leidenuniv.nl/~pbruin/moduli.pdf | |
| Mar 1, 2021 at 10:19 | comment | added | François Brunault | 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. | |
| Feb 28, 2021 at 23:07 | comment | added | k.j. | @FrançoisBrunault Thank you. But it seems that the convergences do not work well. The curve $E(q)$ you mention is the 1-gon when $q = 0$. But why does this mean, on $X_1(N)$, that $E(q)$ (with an approapriate structure) $ \in X_1(N)(\mathbb{C})$ converges to the 1-gon? What topology should I consider? | |
| Feb 28, 2021 at 8:19 | comment | added | François Brunault | 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 $\mathbb{C}((q^{1/N}))$. | |
| Feb 26, 2021 at 21:41 | comment | added | k.j. | @FrançoisBrunault In it, it seems for me that the authors show only the bijectivity, and not the holomorphy. In the 3rd paragraph of section 9.3, the authors says that the bijection is holomorphic, but I couldn't find its proof. | |
| Feb 26, 2021 at 21:31 | comment | added | François Brunault | In the beginning of section 9.3 you can find the discussion over $\mathbb{C}$. Then they go on to define the moduli space $\mathcal{X}_1(N)$ over $\mathbb{Z}[1/N]$ and you can work out the Galois action from the moduli interpretation. | |
| Feb 26, 2021 at 17:18 | comment | added | k.j. | @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. | |
| Feb 26, 2021 at 13:21 | comment | added | François Brunault | This is explained in Diamond-Im, Modular forms and modular curves, section 9.3. You should get the Galois action from that. | |
| Feb 26, 2021 at 12:54 | history | edited | k.j. | CC BY-SA 4.0 |
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| Feb 26, 2021 at 12:44 | history | asked | k.j. | CC BY-SA 4.0 |