Timeline for Splitting of small primes in number fields generated by the torsion of elliptic curves
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Mar 31, 2021 at 22:52 | vote | accept | Asvin | ||
| Mar 15, 2021 at 16:51 | comment | added | Will Sawin | @2734364041 Sure - it's certainly worthwhile to think about what we might be able to prove using $L$-functions. I just think your first comment was misleading in that it implies we have reason to think that the smallest split prime is greater than $\log (D_K)$, when in fact we don't have reason to think this. (We have reason to think that we won't be able to disprove it, but that's different.) | |
| Mar 15, 2021 at 16:30 | comment | added | 2734364041 | When I say "Chebotarev type arguments", I mean "what do zeros of Hecke/Artin $L$-functions indicate?" In certain situations (like splitting conditions in ring class fields / representation of primes by BQFs), you can use geometry or other "low-tech" means to handle the problem, and the zeros are incapable of detecting these super-small split primes. I'm not trying to suggest that the zeros handle everything. I'm suggesting that in this problem, the conductor of the elliptic curve might come into play. Maybe it won't. But it's something to think about, that's all. | |
| Mar 15, 2021 at 15:51 | comment | added | Will Sawin | @2734364041 Fiori's example is for a very specific type of extension. Saying it "has size in the range" implies that the lower bound holds for all extensions. | |
| Mar 15, 2021 at 14:48 | comment | added | 2734364041 | It depends on the Galois extension. See Fiori's "Lower bounds for the least prime in Chebotarev" for examples where least split prime is naturally bounded in terms of the discriminant and Sardari's "The least prime number represented by a binary quadratic form" for examples where the least split prime is expected to be bounded naturally in terms of the degree. | |
| Mar 15, 2021 at 13:22 | comment | added | Will Sawin | @2734364041 Shouldn't the lower bound of your range have more to do with the degree of the field than the size of the discriminant? It's not like quadratic fields of discriminant $D$ have no split primes smaller than $\log D$ for $D$ large, for example. | |
| Mar 15, 2021 at 10:26 | comment | added | 2734364041 | Let $K$ be the field generated by the $d$-torsion points of an elliptic curve $E/\mathbb{Q}$; assume for discussion that $E/\mathbb{Q}$ does not have complex multiplication. Chebotarev type arguments would optimistically suggest that the least unramified prime $p$ that splits completely in $K$ has size in the range $\log D_K$ up to $(\log D_K)^2$, where $D_K$ is the absolute discriminant of $K$. To go further than that, you would need to use other ideas. Note that each prime dividing $D_K$ also divides $dN$, and conversely. The power to which each of these primes is raised is another issue. | |
| Mar 14, 2021 at 22:51 | history | answered | Will Sawin | CC BY-SA 4.0 |