Timeline for What arithmetic information is determined by the $j$-invariant of an elliptic curve?
Current License: CC BY-SA 4.0
10 events
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| Jul 20, 2019 at 19:26 | comment | added | Stanley Yao Xiao | @JoeSilverman perhaps it is better I send you an email explaining the exact question I am interested in, as I think the comment box is not large enough for that | |
| Jul 19, 2019 at 18:10 | comment | added | Joe Silverman | Which phenomena? The $p$-adic fact is standard, the number of components on the Neron model is the valuation of the discriminant. The height result probably isn't written down anywhere, and I'm not 100% sure it's correct; but if it is, it should follow from the methods in my 1981 Duke Math J paper, since one can pigeon-hole to get $\hat h(P)$ larger than a multiple of $\log|j|$, and if $|j|$ is big enough, that will overwhelm the worst possible contribution of $-\frac1{24}\log\Delta$ coming from the finite places. If you have an application ,I could work it out more carefully. | |
| Jul 19, 2019 at 17:27 | comment | added | Stanley Yao Xiao | @JoeSilverman everything you wrote in your comment is very interesting to me... do you have references for the phenomena you mentioned? | |
| Jul 19, 2019 at 15:16 | comment | added | Joe Silverman | Roughly speaking, large $|j|_p$ means that $E$ has "highly" multiplicative reduction at $p$ (and thus it's Neron polygon has lots of sides). So large $|j|_\infty$ means that $E$ has "highly multiplicative reduction at the infinite place of $\mathbb Q$." One can feed that into various machines to get information. For example, Lang's height lower bound conjecture $\hat h(P)\gg\log|\Delta|$ is true, I think. Is that the sort of thing that you had in mind? | |
| Jul 19, 2019 at 11:03 | history | edited | Stanley Yao Xiao | CC BY-SA 4.0 |
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| Jul 19, 2019 at 11:02 | comment | added | Stanley Yao Xiao | @ChrisWuthrich you are right of course; I had this in mind when I insisted that $\gcd(A,B) = 1$, but ultimately the wording of the question did not make this clear. I will edit the question | |
| Jul 19, 2019 at 8:23 | comment | added | Chris Wuthrich | In the $p$-adic version, large $\vert j(E)\vert _p$ measures if $E$ has potential multiplicative reduction and how many components it has. | |
| Jul 19, 2019 at 8:21 | comment | added | Chris Wuthrich | It is more cautious to talk about what arithmetic information one can gain for the family of (quadratic) twists of elliptic curves $E/\mathbb{Q}$ with that $j$-invariant. Or do you choose one curve in that family? | |
| Jul 19, 2019 at 5:07 | comment | added | reuns | Why not choosing $y^2= 4x^3-27j/(j-1728)x-27j/(j-1728)$, the arithmetic complexity is about how it differs from the integer equation with minimal discriminant | |
| Jul 19, 2019 at 0:56 | history | asked | Stanley Yao Xiao | CC BY-SA 4.0 |