## Where can I find a comprehensive list of equations for small genus modular curves?

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Does there exist anywhere a comprehensive list of small genus modular curves $X_G$, for G a subgroup of GL(2,Z/(n))\$? (say genus <= 2), together with equations? I'm particularly interested in genus one cases, and moreso in split/non-split cartan, with or without normalizers.

Ken Mcmurdy has a list here for $X_0(N)$, and Burcu Baran writes down equations for all $X_{ns}^+(p)$ of genus <=2 in this preprint.

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## 5 Answers

No, there does not exist a comprehensive list of equations: the known equations are spread out over several papers, and some people (e.g. Noam Elkies, John Voight; and even me) know equations which have not been published anywhere.

When I have more time, I will give bibliographic data for some of the papers which give lists of some of these equations. Some names of the relevant authors: Ogg, Elkies, Gonzalez, Reichert.

In my opinion, it would be a very worthy service to the number theory community to create an electronic source for information on modular curves (including Shimura curves) of low genus, including genus formulas, gonality, automorphism groups, explicit defining equations...In my absolutely expert opinion (that is, I make and use such computations in my own work, but am not an especially good computational number theorist: i.e., even I can do these calculations, so I know they're not so hard), this is a doable and even rather modest project compared to some related things that are already out there, e.g. William Stein's modular forms databases and John Voight's quaternion algebra packages.

It is possible that it is a little too easy for our own good, i.e., there is the sense that you should just do it yourself. But I think that by current standards of what should be communal mathematical knowledge, this is a big waste of a lot of people's time. E.g., by coincidence I just spoke to one of my students, J. Stankewicz, who has spent some time implementing software to enumerate all full Atkin-Lehner quotients of semistable Shimura curves (over Q) with bounded genus. I assigned him this little project on the grounds that it would be nice to have such information, and I think he's learned something from it, but the truth is that there are people who probably already have code to do exactly this and I sort of regret that he's spent so much time reinventing this particular wheel. (Yes, he reads MO, and yes, this is sort of an apology on my behalf.)

Maybe this is a good topic for the coming SAGE days at MSRI?

Addendum: Some references:

Kurihara, Akira On some examples of equations defining Shimura curves and the Mumford uniformization. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 25 (1979), no. 3, 277--300.

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Reichert, Markus A. Explicit determination of nontrivial torsion structures of elliptic curves over quadratic number fields. Math. Comp. 46 (1986), no. 174, 637--658.

http://www.math.uga.edu/~pete/Reichert86.pdf

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Gonzàlez Rovira, Josep Equations of hyperelliptic modular curves. Ann. Inst. Fourier (Grenoble) 41 (1991), no. 4, 779--795.

http://www.math.uga.edu/~pete/Gonzalez.pdf

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Noam Elkies, equations for some hyperelliptic modular curves, early 1990's. [So far as I know, these have never been made publicly available, but if you want to know an equation of a modular curve, try emailing Noam Elkies!]

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Elkies, Noam D. Shimura curve computations. Algorithmic number theory (Portland, OR, 1998), 1--47, Lecture Notes in Comput. Sci., 1423, Springer, Berlin, 1998.

http://arxiv.org/abs/math/0005160

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An algorithm which was used to find explicit defining equations for $X_1(N)$, $N$ prime, can be found in

Pete L. Clark, Patrick K. Corn and the UGA VIGRE Number Theory Group, Computation On Elliptic Curves With Complex Multiplication, preprint.

http://math.uga.edu/~pete/TorsCompv6.pdf

This is just a first pass. I probably have encountered something like 10 more papers on this subject, and I wasn't familiar with some of the papers that others have mentioned.

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Really nice answer.  One question: Do any of the algorithmic methods you mention for computing defining equations work integrally, or only over Q? – Tyler Lawson Feb 4 2010 at 4:44
@TL: good question!  (Sounds familiar, in fact.)  Off the top of my head, I would say that the key issue is which of these algorithms work over a field of positive characteristic, and for which characteristics?  E.g. "my" algorithm (a.k.a.: the most immediately obvious algorithm) for $X_1(N)$ will work verbatim over a field of characteristic not dividing $N$, hence (I'm pretty sure) over $\mathbb{Z}[\frac{1}{N}]$.  Doing things correctly at primes of bad reduction will be much harder in general, I fear, although in some very special cases (e.g. genus 0!) you can work these things out.   – Pete L. Clark Feb 4 2010 at 6:07

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Explicit equations for X_1(N) that have been optimized to reduce both the degree and coefficient sizes are available for N <= 50 at http://math.mit.edu/~drew/X1_curves.txt. These were obtained using the algorithm described in http://arxiv.org/abs/0811.0296.

EDIT: Tables of defining equations for X_1(N) for N <= 189 are now available at http://www-math.mit.edu/~drew/X1_altcurves.html

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There is also the paper by Broker, Lauter and Sutherland "Modular polynomials via isogeny volcanoes" http://arxiv.org/abs/1001.0402 which gives a fast (in practice) algorithm to calculate modular polynomials $\Phi_l(X,Y)$ (this is the polynomial satisfying $\Phi_l(j(z),j(lz)) = 0$ where $j$ is the usual $j$-invariant) for $l$ prime which is a highly-singular model for $X_0(l)$, and other analogous polynomials associated, for example, with the modular function $f$ which generates a function field associated with a congruence subgroup of degree 72 over $\Gamma_0(1)$. Sutherland just spoke here yesterday on this. For example he can calculate $\Phi_l(X,Y)$ for $l$ about 20000. The interesting feature in this algorithm is that he can calculate $\Phi_l$ modulo a small prime without actually calculating it over $\mathbb{Z}$ and reducing.

In the papers by Cummins and Pauli and Yang they essentially do their calculations by using "modular units" (cf. Kubert and Lang) which are explicit functions on $X(N)$ (sometimes with character) for which we know the divisor, and then combining them in various ways and using Riemann-Roch type calculations. The method by Broker, Lauter and Sutherland uses the modular interpretation of $\Phi_l$ in terms of isogenies, in a rather clever way. I feel that eventually this will be the way to go.

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+1: there is lots more good work being done in this area than I knew about. MO strikes again! – Pete L. Clark Feb 4 2010 at 19:39

Cummins and Pauli have calculated generators for the function fields of all congruence subgroups of $\text{PSL}_2(\mathbb{Z})$ of genus $\le 24$ in:

http://www.mathstat.concordia.ca/faculty/cummins/congruence/

I haven't looked at this for a few months but I believe that the companion paper http://www.emis.de/journals/EM/expmath/volumes/12/12.2/pp243_255.pdf discusses the generators. In the meantime there is a paper by Yifan Yang "Defining equations of modular curves" Advances in Mathematics Volume 204, Issue 2, 20 August 2006, Pages 481-508

which gives tables of equations for many modular curves, and discusses a methodology for finding "good" equations (i.e. those with small coefficients and a small number of terms in the defining polynomials)

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These are very nice tables for what they contain, but I didn't see any data about defining equations.  Am I missing something?  (If you will permit a pedantic remark: I can tell you what the generators are for the function field of any integral algebraic curve over $\mathbb{Q}$: $x$ and $y$.  It's the relation that's not so easy...) – Pete L. Clark Feb 4 2010 at 6:10
@VM: In the paper of Cummins and Pauli I don't see any data about equations or function fields (and again, please let me know if I'm missing it).  On the other hand, the paper of Yang seems like a must-read for those interested in the problem.   – Pete L. Clark Feb 4 2010 at 14:43

Galbraith's thesis has a bunch:

http://www.isg.rhul.ac.uk/~sdg/thesis.html

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There is code in Magma packages to do ModularCurveQuotient which is $X_0(N)$ mod Atkin-Lehners, via Galbraith's thesis. The looking at it seems that you can just change ModularForms(N,2) to ModularForms(Gamma1(N),2) in the function internals and hope to work with no Atkin-Lehners. This gives a canonical embedding to $C^{g-1}$ if so. Why you want this for $g=48$ with $X_1(50)$ as 1035 quadrtics is unclear but it ran in 2 minutes. – Junkie May 1 2010 at 1:51
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