Timeline for Distribution of the rank of $y^2=x^4+x+b^2$
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 7, 2021 at 14:42 | comment | added | Will Sawin | How is the finiteness of Sha relevant to equidistribution of the parity? | |
| Feb 16, 2021 at 16:01 | comment | added | joro | Thanks. sage and mwrank are slow in computing the rank for large $b$, is there a CAS which can compute the rank for say 10 b's for $b$ as large as possible? | |
| Feb 16, 2021 at 15:50 | comment | added | Joe Silverman | @joro I'm just suggesting a line of attack that might be worth checking. But there is also definitely a "small number" phenomenon that shows up in experiments involving ranks in families, even with much larger search spaces than you've used. Indeed, there are often error terms like $O(1/\log|b|)$ if the family is parametrized by $b$. So checking $|b|\le1000$ can be mildly suggestive, but that $27\%$ you get could become (say) $15\%$ if you went to $|b|\le10000$ and $9\%$ if you went to $|b|\le100000$, and it would still really not help in guessing whether the percentage goes to~$0$. | |
| Feb 16, 2021 at 14:41 | comment | added | joro | @JoeSilverman Experimentally we have rank 5 with probability about 27%. Are you trying to explain this with rational function equal square with probability about 27%? | |
| Feb 16, 2021 at 2:22 | comment | added | Joe Silverman | A more likely reason for the rank $5+$ curves is that there is some rational function $f(c)\in\mathbb Q(c)$ such that $$E_{f(c)}:y^2+2y=x^3-4f(c)^2x$$ has rank 4 over $\mathbb Q(c)$. In more geometric language, $E_b\to\mathbb P^1_{\mathbb Q}$ is an elliptic surface of rank 3, and maybe there is a finite cover $\mathbb P^1_{\mathbb Q}\to \mathbb P^1_{\mathbb Q}$ so that the pull-back elliptic surface acquires one or more new independent sections. One way to search for $f(c)$ is to assume it has small degree, use the $b_i$ values with high rank, and solve $f(c)=b_i$ for the coeffs of $f$. | |
| Feb 15, 2021 at 22:04 | history | answered | Daniel Hast | CC BY-SA 4.0 |