Timeline for Is every sufficiently general monic quartic rational square infinitely often?
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Feb 1, 2021 at 14:05 | comment | added | Will Sawin | @Wojowu I just mean any elliptic curve defined by an equation whose coefficients are polynomials in some variables $x_1,\dots,x_n$, as you specialize to different rational (or integer) values of the variables $x_1,\dots,x_n$. Alternately, you can look at a morphism $\mathcal E \to X$ where $X$ is a Fano variety and the generic fiber is an elliptic curve. | |
| Feb 1, 2021 at 14:00 | comment | added | Wojowu | Thank you for pointing out the flaw in my (naive) reasoning. Could you explain what is exactly meant with "geometric family of elliptic curves" here? | |
| Jan 30, 2021 at 16:35 | comment | added | Will Sawin | In my $z, w$ notation, the power series is $z = \pm ( 1 + (b_3/2) w + (b_2/2- b_3^2/8)w^2 + \dots)$. If you plug that in you should be able to clear a power of $w$ (depending on $+$ or $-$ ) | |
| Jan 30, 2021 at 16:33 | comment | added | Will Sawin | @joro I can't read your code very well, but to avoid division by zero you can either use the equation to manipulate the rational functions so there is not a zero in both the numerator and denominator, or else write the first few coefficients of a solution of the equation in bower series around $(0,1)$ or $(0,-1)$, plug the power series into your formula, and clear a power of the variable. One of them should map to the point at infinity, the other to a finite point. | |
| Jan 30, 2021 at 15:53 | comment | added | joro | Thanks. I can't map neither (0,1) nor (0,-1) to the Weierstrass form because of division by zero. | |
| Jan 30, 2021 at 15:15 | history | answered | Will Sawin | CC BY-SA 4.0 |