Timeline for The exponent of Ш of $y^2 = x^3 + px$, where $p$ is a Fermat prime
Current License: CC BY-SA 3.0
19 events
| when toggle format | what | by | license | comment | |
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| S Sep 8, 2020 at 17:03 | history | bounty ended | CommunityBot | ||
| S Sep 8, 2020 at 17:03 | history | notice removed | CommunityBot | ||
| S Aug 31, 2020 at 15:48 | history | bounty started | R.P. | ||
| S Aug 31, 2020 at 15:48 | history | notice added | R.P. | Draw attention | |
| Dec 20, 2016 at 21:30 | history | protected | Felipe Voloch | ||
| S Oct 12, 2016 at 21:22 | history | suggested | C.F.G | CC BY-SA 3.0 |
improve formatting
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| Oct 12, 2016 at 21:21 | review | Suggested edits | |||
| S Oct 12, 2016 at 21:22 | |||||
| Jan 19, 2015 at 22:21 | history | edited | R.P. | CC BY-SA 3.0 |
change from "\colon" to actual colon
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| Nov 3, 2014 at 13:20 | comment | added | joro |
Magma returns some information about 2^32+1. Try the commands: d:=2^(2^5)+1;E := EllipticCurve([0, 0, 0, d, 0]);MordellWeilShaInformation(E);. Returns (Z/2)^4 <= Sha(E)[4] <= (Z/2)^4 and <4, [ 0, 0 ]>
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| Dec 13, 2013 at 10:26 | comment | added | Chris Wuthrich | I don't think that it is easy with modular symbols. If it were a quadratic twist, then there is a formula for all twists with only one modular symbol, but here the modular symbols are maybe not directly related as we have (non-abelian) quartic twists. But there are formulae for $L$-values for these curves as $d$ varies. | |
| Dec 13, 2013 at 1:49 | history | edited | R.P. | CC BY-SA 3.0 |
errors in the statement of the result from Silverman
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| Dec 12, 2013 at 17:58 | comment | added | R.P. | Chris and Tim: thank you both very much. I am going to look into that next. | |
| Dec 12, 2013 at 17:56 | comment | added | Tim Dokchitser | I think Chris is right: your statement is equivalent to $L(E_d,1)/\Omega=2,8,32$ for $d=17,257,65537$, and this quotient could be perhaps computed with modular symbols? (Or at least proving the congruence $L(E_p,1)/\Omega\cong 0$ mod $2^{...}$ using that $p=2^{2^k}+1$?) | |
| Dec 10, 2013 at 22:22 | comment | added | Chris Wuthrich | The value of the $L$-function at $s=1$ for (quartic) twists of $y^2 = x^3+x$ could help. Or maybe the $2$-adic $L$-function. But I don't know enough about CM-curves to help you. | |
| Dec 10, 2013 at 21:42 | history | edited | R.P. | CC BY-SA 3.0 |
added 859 characters in body; edited title
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| Dec 10, 2013 at 17:23 | comment | added | R.P. | Indeed I was wondering about that. Is there a particular reference for this that you would recommend? | |
| Dec 10, 2013 at 17:21 | comment | added | Franz Lemmermeyer | There are similar phenomena in connection with class groups of quadratic fields whose discriminants are Mersenne or Fermat primes. These things are not trivial. | |
| Dec 10, 2013 at 17:06 | history | edited | R.P. | CC BY-SA 3.0 |
fixed grammar in a sentence
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| Dec 10, 2013 at 16:31 | history | asked | R.P. | CC BY-SA 3.0 |