Timeline for Rational Isogenies of Prime Degree
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| Nov 3, 2021 at 15:26 | comment | added | Maarten Derickx | The results in this answer have been published in the meantime doi.org/10.1017/S1474748013000182 | |
| Jan 8, 2011 at 2:50 | comment | added | Dmitry Vaintrob | Sorry not to have got back to you sooner. Hopefully our paper will be up on arXiv soon, but so far I can give you the bounds. Conditionally (on the GRH) they're as follows: Suppose K is a field not containing the Hilbert class field of an imaginary quadratic field. Set S_K the set of primes $\ell$ such that there exists an $\ell$-isogeny. Then the product of all primes in S_K is bounded by something on the order of $\exp((12n)^n(R+h^2\log^2 \Delta)+n^2 h^2 \log^2\Delta)$ where $n$, $\Delta$, $h$, $R$ are the degree, discriminant, class number and regulator of $K$. | |
| Oct 22, 2010 at 9:39 | vote | accept | Barinder Banwait | ||
| Oct 19, 2010 at 13:53 | comment | added | Barinder Banwait | Thanks for your post. I'm very interested in your result. Regarding your first paragraph, Momose's paper (see the answer by stankewicz) proves the quadratic case of your result. Have you used different ideas to the isogeny character and its 'rigidity' approach (Theorem A of loc. cit.)? Are your methods effective, and if so, are the bounds sharp or reasonable? In the quadratic case, how do your bounds compare with Momose's bounds? Regarding the imaginary quadratic case, about all but finitely many isogeny primes must split, this was known to Mazur (Proposition 8.1 in [1]) | |
| Oct 18, 2010 at 23:49 | history | edited | Dmitry Vaintrob | CC BY-SA 2.5 |
Shortened the answer, and (hopefully) made it clearer that this is not yet a finished result.; added 2 characters in body
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| Oct 16, 2010 at 18:56 | history | edited | Dmitry Vaintrob | CC BY-SA 2.5 |
correction: number of quadratic imaginary fields with class number 1 is known to be finite (not a hypothesis.)
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| Oct 16, 2010 at 17:19 | history | answered | Dmitry Vaintrob | CC BY-SA 2.5 |