Timeline for Is there a "classical" proof of this $j$-value congruence?
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15 events
| when toggle format | what | by | license | comment | |
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| Jun 5, 2011 at 0:23 | answer | added | Noam D. Elkies | timeline score: 49 | |
| Nov 20, 2010 at 3:22 | history | bounty ended | BCnrd | ||
| Nov 19, 2010 at 15:27 | comment | added | BCnrd | Dear David: it is in hand-written form, and will likely remain as such until the intended application (by the person who originally asked me to prove this congruence) bears fruit. | |
| Nov 19, 2010 at 5:43 | comment | added | David Hansen | By the way, is your fancy deformation-theoretic proof of this congruence available in preprint form somewhere? | |
| Nov 18, 2010 at 9:45 | comment | added | Franz Lemmermeyer | Showing that some arithmetic object depending on $n$ only depends on the residue class of $n$ is a reciprocity law; Artin's rec. law, for example, claims that the Frobenius of P only depends on the ideal class of the prime P. So the task is to link the congruence to the action of a Frobenius on a suitable space. I have no idea how to achieve this, however. | |
| Nov 18, 2010 at 6:17 | history | edited | BCnrd | CC BY-SA 2.5 |
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| Nov 13, 2010 at 13:43 | comment | added | Felipe Voloch | I recall that the paper has some p-adic results as well as some mod p^2 results. It might not answer your question but it does more than what is in MR. | |
| Nov 13, 2010 at 4:26 | comment | added | BCnrd | Dear Felipe: I have not seen that paper, so I just looked at the review on MathSciNet. It addresses a formula for the norm (from $\mathbf{F}_{p^2}$ down to $\mathbf{F}_p$) of a difference of distinct supersingular $j$-values in characteristic $p$ that are moreover distinct from 0 and 1728. If we suppose the method might adapt to handle 0 and 1728 (with more work), for characteristic 3 there is only one $j$-value and anyway it is all about a mod-$p$ congruence. So my gut feeling is that this is unlikely to help in proving a mod-81 congruence for our desired CM $j$-values in char. 0. Oh well! | |
| Nov 13, 2010 at 4:02 | comment | added | Felipe Voloch | Have you looked at: de Shalit, E.: Kronecker's polynomial, supersingular elliptic curves, andp-adic periods of modular curves. In: Proceedings of the workshop onp-adic monodromy and the Birch-Swinnerton-Dyer conjecture (Boston 1991). Contemporary Mathematics165, AMS (1994), 135-148 ? | |
| Nov 13, 2010 at 3:00 | history | bounty started | BCnrd | ||
| Nov 11, 2010 at 6:17 | history | made wiki | Post Made Community Wiki by BCnrd | ||
| Nov 11, 2010 at 6:14 | comment | added | BCnrd | David, let me explain why that kind of approach falls crucially short. For the setup you mention, $3$ is unramified in the relevant ring class field, so the $j$-congruence ultimately stems from the fact that the reduction in char. 3 is supersingular with half size of aut. group equal to 6. The comparable result one would obtain in this way for the question posed is that $j(n \zeta) \equiv 0 \bmod 27$ (since 3 is now quadratically ramified in the relevant ring class field, so $3^3$ is really an order 6 congruence for the relevant uniformizer, akin to $3^6$ in your comment). So too weak, alas. | |
| Nov 11, 2010 at 5:14 | comment | added | David Hansen | The Gross-Zagier paper "On singular moduli" proves that $j(\tau) \equiv 2^6 3^3 \mathrm{mod} 3^6$ when $j(\tau)$ comes from an elliptic curve with CM by $\mathscr{O}_{\sqrt{-d}}$ for $-d$ a fundamental discriminant with $d\equiv 1\; \mathrm{mod}\; 3$. Perhaps their technique generalizes to answer your question? | |
| Nov 11, 2010 at 3:21 | history | edited | BCnrd | CC BY-SA 2.5 |
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| Nov 11, 2010 at 2:37 | history | asked | BCnrd | CC BY-SA 2.5 |