The main paper is: <a href=http://arxiv.org/abs/1307.5719>Gonality of the modular curve X1(N)</a> (to appear in J. Algebra).
Most files in this folder are data accompanying this paper.
</p>
Preprint <a href=http://arxiv.org/abs/1202.4355>Low Degree Places on the Modular Curve X1(N)</a> (not submitted) is about the file
<a href=LowDegreePlaces>LowDegreePlaces</a> in this folder.
</p>
We used LLL to search the Z-module of modular units mod constants. The file <a href=gonality>gonality</a> lists (for N=4..300) a modular unit (and its degree)
of the lowest degree that we could find.  This degree is an upper bound for min(degree(non-constant modular units)),
which in turn is an upper bound for the Q-gonality.  The Magma files in the folder <a href=LowerBoundsGonality>LowerBoundsGonality</a> prove
that this upper bound for the gonality is sharp for N <= 40.  Thus, the file <a href=gonality>gonality</a> gives the exact value of the Q-gonality of X1(N)
for N <= 40, and gives upper bounds for N <= 300.
</p>
The file <a href=Subfields>Subfields</a> looks at all decompositions of the map X1(N) --> X1(1) to see if that helps to find a function of lower degree
(none of the several methods we looked at produced a function of a degree that is lower than that found with the LLL method).
</p>
To LLL-search the modular units, we need a basis (see Section 2) and their divisors. These divisors are listed in the
files <a href=cusp_divisors>cusp_divisors</a> and <a href=cusp_divisors_large>cusp_divisors_large</a>.  To interpret these divisors (e.g. to compute
the degree of the corresponding function), you also need the degree of each place, these are listed in <a href=cusp_degrees>cusp_degrees</a>.
</p>
Instead of downloading a file with divisors, you can also download a program that computes these divisors, in the file
<a href=cusp_divisors_program>cusp_divisors_program</a>.  Note that this program uses a different basis
of modular units.  The program uses the modular units F_k in Section 2 of the paper (for k>3,  F_k is the modular equation in b,c coordinates)
while the files <a href=cusp_divisors>cusp_divisors</a> and <a href=cusp_divisors_large>cusp_divisors_large</a> use the modular
units f_k in Section 2.1 (for k>9, f_k is the modular equation in x,y coordinates) (the advantage of x,y coordinates is that they
give smaller expressions than b,c coordinates).