Timeline for modular forms, invertible sheaves, and quotients
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 11, 2014 at 21:35 | vote | accept | Nadim Rustom | ||
| Aug 11, 2014 at 9:42 | comment | added | eric | @Nadim: if we fix distinct points $P$ and $Q$ on projective 1-space $X$, and then we look at the sheaf of functions on $X$ that are holomorphic away from $P$ and $Q$, vanish at $P$ and are allowed a pole at $Q$, then that sheaf is isomorphic to the structure sheaf, and has no base points, but all the sections vanish at $P$ when considered as functions. Is this the point that's confusing you? | |
| Aug 11, 2014 at 9:21 | comment | added | Nadim Rustom | Another question, if $\mathcal{G}_k$ is invertible, what is its degree? (I don't see how it could be an integer). | |
| Aug 11, 2014 at 9:20 | comment | added | Nadim Rustom | Do you mean $\mathcal{G}_6$ and $E_6$? I couldn't think of one, but I'm still not convinced. Are you saying that $E_6$ does not vanish as a section of $\mathcal{G}_6$ at $[i]$ ? (Where then does it vanish?) I find that hard to believe. One can forget about modular forms as functions on the upper-half plane, and look at the moduli interpretation. The elliptic curve corresponding to $j = 1728$ is $y^2 = x^3 + x$, hence $E_6$ must vanish at $j = 1728$. Since any modular form of weight $k \equiv 2 \pmod{4}$ is a multiple of $E_6$, it must also vanish at $j = 1728$. | |
| Aug 10, 2014 at 19:44 | comment | added | David Loeffler | Can you exhibit a section of $\mathcal{G}_4$ on a neighbourhood of $[i] \in X(1)$ which is not a regular function times $E_4$? | |
| Aug 10, 2014 at 12:20 | comment | added | Nadim Rustom | Also, I thought that for any invertible sheaf on a curve the two things are equivalent: an invertible sheaf is generated by global sections if and only if at every point there is a global section not vanishing at that point. | |
| Aug 10, 2014 at 12:19 | comment | added | Nadim Rustom | On the stack $[\Gamma]$ an invertible sheaf $\omega$ exists, I'm trying to look at the pushforward of $\omega^{\otimes k}$ to the coarse moduli curve $X(\Gamma)$, this should be the same as the sheaves $\mathcal{G}_k$. I'm wondering whether this pushforward is locally free on $X(\Gamma)$. I don't quite understand your second sentence. How is $E_4$ a local basis for $\mathcal{G}_k$ if $k \equiv 2 \pmod{4}$ ? | |
| Aug 10, 2014 at 12:18 | comment | added | Nadim Rustom | Dear David, thanks for your reply. if you check out Eyal Goren's book on Hilbert modular forms: books.google.dk/… He says on page 24 that these sheaves whose global sections are modular forms are generally not invertible. His argument is basically the same as mine. | |
| Aug 10, 2014 at 7:57 | history | answered | David Loeffler | CC BY-SA 3.0 |