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 up-to-date.)

• 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 X₀(N), Tokyo J. Math., 18, no. 2, p. 443-456 (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. 779-795 (1991)
  • N. Murabayashi, On normal forms of modular curves of genus 2, Osaka J. Math, 29, p. 405-462 (1992)
  • Yuji Hasegawa, Table of quotient curves of modular curves X₀(N) with genus 2, Proc. Japan Acad. Ser. A Math. Sci. 71, no. 10, p. 235-239 (1995)
  Different methods for obtaining some equations for hyperelliptic modular curves (and more generally, curves whose Jacobian is a factor of J₀(N)) were given by:
  • X. D. Wang, 2-dimensional simple factors of J₀(N), Manuscripta Mathematica, Vol. 87, No. 2, p. 179-197 (1995)
  • Hermann-Josef Weber, Hyperelliptic simple factors of J₀(N) with dimension at least 3, Experimental Mathematics, Vol. 6, No. 3, p. 235-249 (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 Q-curves, related to that of Chapter 6 of the thesis. The connection between Q-curves and rational points on quotients of modular curves was noted by Elkies:
  • Noam Elkies, Remarks on elliptic K-curves, preprint (1993)
  Examples of j-invariants of Q-curves 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, Q-curves over quadratic fields, Manuscripta Math., 94, p. 347-364 (1997)
  • Josep Gonzalez and J.-C. Lario, Rational and elliptic parameterisations of Q-curves, J. Num. Th., 72, p. 13-31 (1998)
  Data on the j-invariants of the quadratic Q-curves provided by the rational points found in Chapter 6 of the thesis is given in:
  • Steven Galbraith, Rational points on X₀⁺(p), Experimental Math., 8, No. 4, p. 311-318 (1999)
  More information on the j-invariants of quadratic Q-curves can be found in:
  • Josep Gonzalez, On the j-invariants of the quadratic Q-curves, Preprint (1998)
  Two new examples of j-invariants of Q-curves (from the case where N is composite) are given in:
  • Steven Galbraith, Rational points on X₀⁺(N) and quadratic Q-curves, gzipped ps.
• The outcome of this work is the following fact:
  Suppose that the genus of X₀⁺(N) is less than or equal to 5. Then X₀⁺(N)(Q) has exceptional rational points (i.e., non-cusp and non-CM) when:
  X₀⁺(N) has genus zero,
  X₀⁺(N) has genus one and N = 37, 43, 53, 61, 65, 79, 83, 89, 101 and 131 (all rank 1),
  X₀⁺(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 X₀⁺(N) is less than or equal to 5

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Last Modified: 18-10-2001