Thesis

Doctoral thesis Equations for modular curves, Oxford 1996.

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.

• 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 William Stein's tables.
• 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