Timeline for Fixed $a_p=p+1-\#E(\mathbb{F}_p)$ and $a_p \ne 0$ on an elliptic curve infinitely often for fixed curve over the rationals?
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Jun 28 at 11:26 | comment | added | joro | @StanleyYaoXiao Thanks. Did the paper prove that there are no other sporadic solutions except primes of certain form? | |
| Jun 27 at 19:48 | comment | added | Stanley Yao Xiao | Looking at the paper, it seems that they rely on the fact that such extremal primes correspond to Gaussian primes in certain narrow sectors, which is can be handled because of the strong equidistribution of roots of quadratic congruences. The problem you ask seems to be roughly equivalent to Landau's conjecture on prime values of $n^2 + k$ for a fixed integer $k$. The latter is basically impossible with current technology. | |
| Jun 27 at 15:25 | comment | added | Stanley Yao Xiao | This question seems related to the question of extremal primes. For CM curves the exact distribution is proved by James (RIP) and Pollack, see sciencedirect.com/science/article/pii/S0022314X16302670 | |
| Jun 27 at 15:10 | answer | added | Will Sawin | timeline score: 6 | |
| Jun 27 at 14:23 | history | asked | joro | CC BY-SA 4.0 |