Timeline for Explicit computations of finite covers of genus one curves with two points of ramification
Current License: CC BY-SA 4.0
23 events
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| Apr 19, 2022 at 1:44 | comment | added | Lev Borisov | Thank you! I will first try Will Sawin's example, since I am comfortable with modular forms, but will come back to Will Chen's and your suggestions. | |
| Apr 18, 2022 at 20:59 | comment | added | Jason Starr | It may be difficult to write explicit equations for these covers. Instead one can start with the genus $0$, degree $7$ cover branched over $3$ points $\{p,q,r\}$ with branching of type $(7)$, of type $(2,2)$, and of type $(3,3)$. This is an intermediate cover of the Belyi map from the Klein quartic, and so can be made explicit. Now take a genus $1$, degree $6$ cover of the projective line branched over $\{p,q,r,s\}$ with cycle types $(3,3)$, $(2,2,2)$, $(3,3)$, and $(2)$, i.e., simple branching at $s$. Of course now the difficulty is explicitly describing this degree $6$ cover. | |
| Apr 18, 2022 at 20:54 | comment | added | Jason Starr | I am just adding some more details. The cover is the genus $2$, degree $7$ branched cover of the projective line branched over $4$ points, $\{p,q,r,s\}$, where over two of the points, say $p$ and $q$, the cycle types are $7$-cycles, and over the other two points the cycle types are $(2,2)$, and these generate $\textbf{PSL}_2(\mathbb{Z}/7\mathbb{Z})$, e.g., the $7$-cycles are $(1234567)$ and its inverse, and the $(2,2)$-cycles are both $(26)(34)$. The genus $1$, degree $2$ cover should be branched over all of $p,q,r,s$. The inverse images of $p$ and $q$ differ by $2$-torsion. | |
| Apr 18, 2022 at 17:45 | comment | added | Lev Borisov | @WillChen Yes, definitely I should check that. I have seen some of your work, but not in detail, so if this turns out to be a $2$-torsion then it would definitely give me a reason to look further. | |
| Apr 18, 2022 at 17:42 | comment | added | Will Chen | ... As a shameless plug, the moduli of PSL(2,p)-covers of elliptic curves only ramified over the origin with ram. index p is interestingly connected with the $\mathbb{F}_7$-points of the Markoff equation. See this and that. | |
| Apr 18, 2022 at 17:38 | comment | added | Will Chen | Also, in the off chance that $p_1 - p_2$ happens to be 2-torsion, then you can consider obtaining $X$ by pulling back (via a 2-isogeny) a degree 7 cover of an elliptic curve only branched over the origin (with ram. index 7). The latter such covers have a remarkably simple moduli space -- it is a genus 0 "noncongruence modular curve", with degree 7 over the $j$-line, corresponding to an index 7 subgroup of SL(2,Z) (technically this is only correct for Galois PSL(2,7)-covers of elliptic curves only ramified over the origin with ram. index 7, but IIRC the moduli spaces should be very similar)... | |
| Apr 18, 2022 at 17:29 | comment | added | Lev Borisov | @WillChen Interesting reference, thank you! My $E$ is fairly ugly, with the $J$-invariant $(2661975021951 - 1086757026275 \sqrt{-7})/4598530048$, so computing Belyi maps could be computationally prohibitive. But it is definitely something to think about. | |
| Apr 18, 2022 at 17:23 | comment | added | Will Chen | ...this would at least give a potential algorithmic solution to your problem (assuming by "explicit" you mean that you want equations for $X,f$), but I don't know how enlightening this would be for you. Doing this efficiently is still a bit of an art, but I think John Voight has assembled a "pipeline" for computing Belyi maps for degrees which aren't too big ($\le 300$ is probably fine). You could consider reaching out to him if you need explicit equations. | |
| Apr 18, 2022 at 17:13 | comment | added | Will Chen | You could also phrase everything in terms of the problem of computing equations for Belyi maps. For example, you can define a suitable $X$ group theoretically using mondromy data over $E$, then compute a Belyi map of your elliptic curve ramified at $p_1,p_2$, and compose with the cover to get a Belyi map $f : X \rightarrow\mathbb{P}^1$. You can then compute the monodromy description for $f$, and try to find equations for $X,f$ using existing techniques for computing Belyi maps from monodromy data (see, eg, arxiv.org/pdf/1311.2529.pdf)... | |
| Apr 18, 2022 at 2:14 | comment | added | Lev Borisov | One can try to take the divisor $4(\hat p_1 + \hat p_2)$ where $\hat p_i$ are the preimages of $p_i$ which will likely give a map $X\to \mathbb P^2$, hopefully with image a nodal octic curve. The coefficients of the equation of the octic should be given by some algebraic functions in the two parameters of $M_{g=1,n=2}$, so if these can be understood to sufficiently high order near a specific point, then one should be able to find them. | |
| Apr 18, 2022 at 1:09 | comment | added | Lev Borisov | I would deform to high order step by step and hopefully get something. Not a sure thing, but worth a shot. Thank you! | |
| Apr 18, 2022 at 0:42 | comment | added | Will Sawin | It does, but the ramification should all be canceled by the ramification of $X_0(11) $, which is totally ramified over $i$ and $\rho$ (since $11$ in inert in $\mathbb Q(i)$ and $\mathbb Q(\rho)$). However, I didn't fully process the specific elliptic curve aspect of this, and I'm not very optimistic that deforming the equation will be enlightening. But I don't disagree it's worth a shot... | |
| Apr 18, 2022 at 0:37 | comment | added | Lev Borisov | @WillSawin It's an interesting idea and I at first thought it might work, but now I think that the map $X(7)\to X(1)$ has other ramification points. So I am worried about additional ramification over $i$ and $\rho$. But it's easy to check! | |
| Apr 18, 2022 at 0:36 | comment | added | Jason Starr | Actually the genus $1$ cover has to have all four branch points equal to the four branch points of the genus 2 cover of degree $7$. | |
| Apr 18, 2022 at 0:17 | comment | added | Lev Borisov | @JasonStarr What is this cover? Is there a reference? | |
| Apr 18, 2022 at 0:04 | comment | added | Will Sawin | One approach would be to calculate explicit equations for the modular curve $X_0(11) \times_{X(1)} X(7)$ as a cover of $X_0(11)$ using modular forms methods. It's not so hard to check that this cover has all the properties you want. You can replace $X_0(11)$ by any modular curve of genus $1$, with level prime to $7$, with no elliptic points and two cusps. | |
| Apr 18, 2022 at 0:04 | comment | added | Jason Starr | There is a degree $7$ cover of the projective line with Galois group $\textbf{PSL}_2(\mathbb{Z}/7\mathbb{Z})$ with $2$ branch points with branching equal to a $7$-cycle and with $2$ branch points with branching equal to a product of two $2$-cycles. Pullback this cover by any degree $2$ cover of the projective line by a genus $2$ curve whose four branch points include the two branch points with $(2,2)$-branching. | |
| Apr 17, 2022 at 23:58 | history | edited | Lev Borisov | CC BY-SA 4.0 |
added 165 characters in body
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| Apr 17, 2022 at 23:52 | comment | added | Lev Borisov | Of course it has, it's the stabilizer of a point (or a line) in its $GL_3(2)$ incarnation. What I am asking is how one can go about computing an explicit equation of the cover. | |
| Apr 17, 2022 at 23:50 | comment | added | Jason Starr | Are you asking whether the group $\mathbf{PSL}_2(\mathbb{Z}/7\mathbb{Z})$ has a subgroup of index $7$? | |
| Apr 17, 2022 at 19:53 | comment | added | Lev Borisov | My question is whether there are any similar problems solved in the literature. Perhaps I could use some of their approaches, if such exist. | |
| Apr 17, 2022 at 19:40 | comment | added | Will Chen | What is your question? | |
| Apr 17, 2022 at 18:48 | history | asked | Lev Borisov | CC BY-SA 4.0 |