Timeline for Is there an isotrivial elliptic surface of positive rank having a section of order $3$?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Nov 13, 2023 at 10:24 | comment | added | Dimitri Koshelev | What if an elliptic surface is isotrivial? | |
| Nov 12, 2023 at 20:00 | comment | added | Ariyan Javanpeykar | No. The moduli map (j-invariant) of your elliptic surface has a degree, say $d$. This will also be the degree of the moduli map of your specialized elliptic surface over $K:=\overline{\mathbb{F}_p}$ for almost all prime numbers $p$. So, you can't have more than $d$ points in $\mathbb{P}^1(K)$ with the same $j$-invariant. | |
| Nov 9, 2023 at 20:56 | comment | added | Dimitri Koshelev | The degrees of the coefficients $a_4(g)$, $a_6(g)$ of the induced elliptic surface depend on $q$. For my applications this is bad. Is it possible to construct an elliptic surface over $\mathbb{Q}$ (or some number field) whose reductions at infinitely many primes (or prime powers) $q$ give desired elliptic $\mathbb{F}_{\!q}$-surfaces? | |
| Nov 9, 2023 at 9:36 | comment | added | Ariyan Javanpeykar | Let $k$ be a finite field and let $S$ be a finite set of closed points of $\mathbb{P}^1_{\mathbb{F}_p}$. Choose a function $g:X:=\mathbb{P}^1\to \mathbb{P}^1$=:Y which sends $S$ to $0$. Now, let $\pi:E\to \mathbb{P}^1_{k}$ be an elliptic fibration with a smooth fibre $E_0$ over $0$. Let $j_0$ be its $j$-invariant. Consider the pullback of $\pi$ along $g$. This is an elliptic fibration over $\mathbb{P}^1$ (again) whose fibres over $S$ are the same as those of $\pi$ over $0$. You can pick $S=\mathbb{P}^1(k)$ if you like. Does this answer your question? | |
| Nov 8, 2023 at 14:02 | comment | added | Dimitri Koshelev | Thank you for your detailed answer. What about my more general question about the existence of an elliptic $\mathbb{F}_{\!q}$-surface over a quite large finite field $\mathbb{F}_{\!q}$ with $\mathbb{F}_{\!q}$-isomorphic fibers (maybe, except for a small number of fibers)? This is a more subtle question. | |
| Nov 7, 2023 at 20:22 | comment | added | Ariyan Javanpeykar | Ow, right. I was too quick there. I made a quick edit excluding $j=0$. | |
| Nov 7, 2023 at 20:21 | history | edited | Ariyan Javanpeykar | CC BY-SA 4.0 |
added 208 characters in body
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| Nov 7, 2023 at 19:39 | comment | added | Will Sawin | One additional necessary detail: $Y_1(3)$ does have stacky points, but these points have $j$ invariant $0$, so the assumption that the $j$ invariant is not zero must be used. | |
| Nov 7, 2023 at 14:39 | history | answered | Ariyan Javanpeykar | CC BY-SA 4.0 |