Timeline for An explicit equation of the canonical morphism $X_1(N) \to X_0(N)$
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 16, 2021 at 0:12 | comment | added | k.j. | Thank you very much! I understand that! | |
| May 15, 2021 at 9:02 | comment | added | François Brunault | You may need to adjust the parameter $d$ and the precision of the $q$-expansion (number of rows). For $N=11$, the modular curves $X_1(N)$ and $X_0(N)$ are elliptic curves so there are other methods. The morphism $\pi$ is just $11.a3 \to 11.a2$ in lmfdb.org/EllipticCurve/Q/11/a It is the quotient of $X_1(11)$ by the $5$-torsion subgroup (which is the orbit of $\infty$ under the diamond operators). In this case you can get equations for $\pi$ using Vélu's formulas, it is implemented in Pari/GP with the function ellisogeny. | |
| May 15, 2021 at 8:59 | comment | added | François Brunault | With a computer algebra system like Pari/GP or Sage, it is not difficult. You set up a matrix whose columns are indexed by your functions $F^i X$ and $F^i G^j$ (create a separate vector which remembers how you order the columns). Each column gives the coefficients of the $q$-expansion of the corresponding function (taking into account that there may be negative exponents, so you should consider the minimal valuation of all these functions, at which every column should start). Then compute the (right) kernel of the matrix. | |
| May 15, 2021 at 8:26 | comment | added | k.j. | Thank you. Would you explain it for explicit $X_1(N) \to X_0(N)$, for example $N = 11$? I tried it, but because there are so many terms I can't find the relation well... | |
| May 15, 2021 at 7:35 | vote | accept | k.j. | ||
| May 15, 2021 at 7:30 | comment | added | François Brunault | @k.j. You're welcome. I should have explained it: just compute the $q$-expansions of all the functions $F^i X$ and $F^i G^j$, and then find a non-trivial linear relation between them by solving the linear system over $\mathbb{Q}$. | |
| May 15, 2021 at 7:03 | comment | added | k.j. | Thank you very much! I've read the paper and computed the q expansions of some $F$, $G$, $X = \Sigma_\gamma F |_\gamma$ and similar for $Y$. But how can I find the relation as in (*)? | |
| May 10, 2021 at 20:43 | history | edited | François Brunault | CC BY-SA 4.0 |
Slight additions and clarifications
|
| May 10, 2021 at 15:57 | history | edited | François Brunault | CC BY-SA 4.0 |
Expanded the answer and corrected typos
|
| May 8, 2021 at 20:05 | history | edited | François Brunault | CC BY-SA 4.0 |
added 371 characters in body
|
| May 8, 2021 at 19:45 | history | edited | François Brunault | CC BY-SA 4.0 |
Added explanation on how to certify the relation
|
| May 8, 2021 at 19:31 | history | answered | François Brunault | CC BY-SA 4.0 |