Timeline for Why is there a weight 2 modular form congruent to any modular form
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| Aug 30, 2011 at 23:31 | history | edited | Tommaso Centeleghe | CC BY-SA 3.0 |
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| Jul 10, 2011 at 12:43 | comment | added | Tommaso Centeleghe | @Kevin: thanks, I see now why you spoke of coherent cohomology, modular forms live in degree zero in this context. | |
| Jul 10, 2011 at 11:34 | comment | added | Kevin Buzzard | @Tommaso: I thought that the way to relate forms to functions on the adelic quotient space was via considering a mod $p$ modular form as a section of the coherent sheaf $\omega^k$ on the mod $p$ modular curve, and then restrict these sections to the supersingular locus and analyse a la Edixhoven's paper. | |
| Jul 9, 2011 at 13:41 | comment | added | Tommaso Centeleghe | (This is because the determinant is on the $\ell$-th component of $K_N$ surjective onto $\mathbf{Z}_\ell^*$) A consequence of Serre' thm. is that then for any mod p Eisestein eigensystem $B$ arising from level $\gamma_1(N)$, and different from $(\ell^k+\ell^{k+1})$, there is an infinte dimensional cusp form $\Pi$ on $G$ such that the associated eigensystem reduces mod $p$ to $B$. I know this looks strange (I think once we discussed this issue on MO). | |
| Jul 9, 2011 at 13:41 | comment | added | Tommaso Centeleghe | Kevin thanks for your comments. Why is it the "coherent cohomology" answer? Concerning (2), let me add that the open subgroup $K_N$ that I am using is locally at $\ell\neq p$ given by invertible matrices that are congruent to $(* *; 0 1)$ mod $N$ (and it is max'l pro-p at p). If a function $f:G(A)\rightarrow \mathbf{C}$ factors through the reduced norm $G(A)\rightarrow A^*$, and it is invariant to the right by $K_N$, and to the left by $G(\mathbf{Q})$, then $f$ has to be invariant to the right under $K_1$. | |
| Jul 9, 2011 at 12:55 | comment | added | Kevin Buzzard | Remarks: (1) I gave the "etale cohomology" answer and your answer is the "coherent cohomology" answer -- i.e. it doesn't go via Eichler-Shimura. In fact there's another "coherent cohomology" approach to the question -- explained e.g. in Edixhoven's Inventiones paper from 1990 or so on Serre's conjecture. (2) Minor point: I am not so sure about your claims about when $\Pi_B$ is infinite-dimensional. You have a tame level $\Gamma_1(N)$ and so your character $\chi$ can I think be ramified at these primes too. But all these cases can be dealt with using Eisenstein series anyway without any truble. | |
| Jul 9, 2011 at 12:20 | history | answered | Tommaso Centeleghe | CC BY-SA 3.0 |