Thesis
Doctoral thesis
Equations for modular curves, Oxford 1996.
(gzipped ps)
TYPO: pointed out by
Vishal Arul
in the equation for X_{0}^{+}(157): everything is correct except for the third equation (first two equations and rational points are fine), where the coefficient of w^2 should be 2 instead of 1.
A number of people have been interested in some of the results of this
thesis. The following references contain further information on some
of these topics. (This only lists references up to the late 1990s. This bibliography is not kept uptodate.)
 The use of the canonical embedding to obtain equations for
modular curves as in Chapter 2 (including some of the examples
given in the thesis) is also described in:
 Mahoro Shimura, Defining equations of modular curves X_{0}(N),
Tokyo J. Math., 18, no. 2, p. 443456 (1995)
 Equations for hyperelliptic modular curves as in Chapter 3 have also
been given by:
 Josep Gonzalez, Equations of hyperelliptic modular curves,
Ann. Inst. Fourier, 41, , p. 779795 (1991)
 N. Murabayashi, On normal forms of modular curves of genus 2,
Osaka J. Math, 29, p. 405462 (1992)
 Yuji Hasegawa, Table of quotient curves of modular curves
X_{0}(N) with genus 2,
Proc. Japan Acad. Ser. A Math. Sci. 71, no. 10, p. 235239 (1995)
Different methods for obtaining some equations for
hyperelliptic modular curves (and more generally,
curves whose Jacobian is a factor of J_{0}(N))
were given by:
 X. D. Wang, 2dimensional simple factors of J_{0}(N),
Manuscripta Mathematica, Vol. 87, No. 2, p. 179197 (1995)
 HermannJosef Weber,
Hyperelliptic simple factors of J_{0}(N) with dimension at least 3,
Experimental Mathematics, Vol. 6, No. 3, p. 235249 (1997)
 Gerhard Frey and Michael Mueller,
Arithmetic of modular curves and applications,
in B. H. Matzat (ed.), Algorithmic algebra and number theory, Springer (1999)

To obtain the modular form data for computations such
as these I recommend consulting
LMFDB.
 There has been a lot of work on Qcurves, related to that of
Chapter 6 of the thesis. The connection between Qcurves
and rational points on quotients of modular curves was noted
by Elkies:
 Noam Elkies, Remarks on elliptic Kcurves, preprint (1993)
Examples of jinvariants of Qcurves corresponding to
the cases where the modular curve has genus zero or has
genus one and the Jacobian has positive rank
have been given by:
 Yuji Hasegawa, Qcurves over quadratic fields,
Manuscripta Math., 94, p. 347364 (1997)
 Josep Gonzalez and
J.C. Lario, Rational and elliptic
parameterisations of Qcurves, J. Num. Th., 72, p. 1331 (1998)
Data on the jinvariants of the
quadratic Qcurves provided by the rational points found in
Chapter 6 of the thesis is given in:
 Steven Galbraith, Rational points on X_{0}^{+}(p),
Experimental
Math., 8, No. 4, p. 311318 (1999)
More information on the jinvariants of quadratic Qcurves
can be found in:
 Josep Gonzalez, On the jinvariants of the quadratic Qcurves,
Preprint (1998)
Two new examples of jinvariants of Qcurves (from the case where N is composite) are given in:
 Steven Galbraith, Rational points on X_{0}^{+}(N) and
quadratic Qcurves,
gzipped ps.

The outcome of this work is the following fact:
Suppose that the genus of X_{0}^{+}(N) is less than or equal to 5.
Then X_{0}^{+}(N)(Q) has
exceptional rational points (i.e., noncusp and nonCM) when:
X_{0}^{+}(N) has genus zero,
X_{0}^{+}(N) has genus one and N = 37, 43, 53, 61, 65, 79, 83, 89, 101 and 131 (all rank 1),
X_{0}^{+}(N) has genus between 2 and 5 and N = 73, 91, 103, 125, 137, 191 and 311.
We conjecture that the above cases are the only ones for which
there are exceptional rational points when
the genus of X_{0}^{+}(N) is less than or equal to 5
Back
Last Modified: 18102001