Timeline for What is the Galois group of the modular equation?
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Sep 7, 2019 at 14:51 | comment | added | Zakariae.B | You may check the book of Clemens Adelmann "The decomposition of primes in torsion point fields" ; Section 5.2 contains an interesting answer of your question in a special case. | |
| Sep 7, 2019 at 5:41 | comment | added | Venkataramana | The simplicity of $SL_2({\mathbb Z}/p{\mathbb Z})$ modulo centre for $p\geq 5$ implies that when $N=p$, the Galois group is $PSL_2({\mathbb F}_p)$. | |
| Sep 7, 2019 at 0:32 | answer | added | reuns | timeline score: 1 | |
| Sep 6, 2019 at 19:16 | history | edited | GH from MO |
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| Sep 6, 2019 at 15:54 | comment | added | Shimrod | @NoamD.Elkies, The polynomial $\Phi_n(X,X)$ has integer coefficients, so the question is what is its Galois group over $\mathbf Q$. I would be also interested in the Galois group of $\Phi_n(X,Y)$ over $\mathbf Q(Y)$ and what happens when we specialize the $j$-invariant at a particular CM point. I would really appreciate if you could comment on these questions. | |
| Sep 6, 2019 at 15:37 | comment | added | Noam D. Elkies | Galois group over what field: ${\bf Q}(j(\tau))$ or ${\bf C}(j(\tau))$? It does matter for all but a few $n$. | |
| Sep 6, 2019 at 15:05 | comment | added | Shimrod | @Venkataramana, thanks, corrected. | |
| Sep 6, 2019 at 15:05 | history | edited | Shimrod | CC BY-SA 4.0 |
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| Sep 6, 2019 at 15:04 | comment | added | Venkataramana | Your definition of ${\mathcal M}_n$ does not seem to depend on $n$ | |
| Sep 6, 2019 at 14:30 | comment | added | Shimrod | @Bullet51, we have $j\circ\gamma = j$ whenever $\gamma\in \operatorname {SL}_2(\mathbf Z)$. The product in question is extended over some set of right coset representatives of $\mathcal M_n ^*$ modulo $\operatorname {SL}_2(\mathbf Z)$. | |
| Sep 6, 2019 at 14:10 | comment | added | LeechLattice | Isn't $j(M \tau)=j(\tau)$? | |
| Sep 6, 2019 at 12:30 | history | asked | Shimrod | CC BY-SA 4.0 |