Timeline for Is there a way to define Hecke operators "inherently" as certain endomorphisms of the Jacobian?
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8 events
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| Aug 19, 2010 at 2:18 | comment | added | BCnrd | No. The Hecke operator is useful because relates to mod. forms (and one point of E-S relation is to relate Hecke e-values to Frob e-values on Tate module). The connection between Fourier coeffs of eigenforms and Frobenius e-values is not a matter of clever def'ns, so ad hoc def'n as a lift of endomorphism of the reduction would surely be useless (& baffling how to make such a lift aside from via moduli-theoretic def'n & using that to compute its effect on the special fiber). Method in Stein's answer surely will hit a brick wall; hard work can't be defined away. Moduli method is ubiquitous. | |
| Aug 19, 2010 at 1:16 | comment | added | David Hansen | Hi Phil! Nice question. | |
| Aug 19, 2010 at 1:00 | answer | added | William Stein | timeline score: 3 | |
| Jul 29, 2010 at 15:22 | comment | added | Philip Engel | Optimally, $X$ would be any Riemann surface such that the endomorphism ring of its Jacobian is defined over $\mathbb{Q}$. In this case, the definition of $T_p$ couldn't rely an interpretation as a moduli space or quotient of the upper half-plane. This definition would coincide with the one we know for modular curves. | |
| Jul 29, 2010 at 6:28 | comment | added | Pete L. Clark | You don't say exactly what $X$ is, but from the context it must be something like $X_0(N)$ or $X_1(N)$ -- i.e., a moduli space of elliptic curves. So it would seem to be hard to get away from the moduli interpretation (but it is possible: there are other ways to think about these curves). How are you thinking about $X_0(N)$ if not as a moduli space or as $\Gamma_0(N) \backslash \overline{\mathcal{H}}$? | |
| Jul 29, 2010 at 5:16 | comment | added | algori | ... or in the answer to this question: mathoverflow.net/questions/26871/… | |
| Jul 29, 2010 at 4:51 | comment | added | Qiaochu Yuan | Hi, Philip! You might be interested in some of the answers I got when I asked about the Eichler-Shimura relation: mathoverflow.net/questions/19390/… | |
| Jul 29, 2010 at 4:48 | history | asked | Philip Engel | CC BY-SA 2.5 |