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Unifying themes suggested by Belyi’s Theorem

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Number Theory, Analysis and Geometry

Abstract

Belyi’s Theorem states that every curve defined over the field of algebraic numbers admits a map to the projective line with at most three branch points. This paper describes a unifying framework, reaching across several different areas of mathematics, inside which Belyi’s Theorem can be understood. The paper explains connections between Belyi’s Theorem and (1) The arithmetic and modularity of elliptic curves, (2) abc-type problems and (3) moduli spaces of pointed curves.

Mathematics Subject Classification (2010): Primary 11G99; Secondary 11G32, 11Dxx, 11G

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Notes

  1. 1.

    Apparently one reason for this is that [Bel02] was an MPI preprint for several years that was difficult to access before it was published.

  2. 2.

    This is true so long as m i ≠0 for all i which holds by (2.2.14) and (2.2.18).

  3. 3.

    In other words, regard (2.2.27) as an identity in Q[λ1, …, λ n ].

  4. 4.

    The deep result here, which we do not use, is the Semistable Reduction Theorem of Deligne-Mumford, which says that every curve acquires semistable reduction over some finite extension of the base field (see [DM69]). However this result is the reason why in Theorem 2.8 we say “semistable bad reduction” rather than simply “bad reduction”, which would have been a stronger statement.

  5. 5.

    So called because it is the function field analogue which motivated the famous ‘abc Conjecture’ of Masser and Oesterlé (see [Oes89]).

  6. 6.

    The only downside to [Ser92] is that it was written before Raynaud [Ray94] proved Abhyankar’s conjecture, so it is a bit dated and does not include the most recent developments in the field.

  7. 7.

    If n ≥ g + 1 and P 1, …, P n are distinct points on C, then the divisor \(D = {P}_{1} + \cdots {P}_{n}\) satisfies \(\mathcal{l}(D) =\deg D - g + 1 + \mathcal{l}(K - D) \geq \deg D - g + 1 \geq2\) by Riemann–Roch, so one can find the desired function in \(\mathcal{L}(D) = {H}^{0}(C,{\mathcal{O}}_{C}(D))\).

  8. 8.

    We thank the referee for bringing this reference to our attention.

  9. 9.

    A generalization of Belyi’s theorem for surfaces in a different direction than the one considered here can be found in [Par02]. Since we believe the generalization in [Par02] is less natural and of a more technical nature, we have chosen not to state it here.

  10. 10.

    In this paper abelian surface will always mean algebraic abelian surface.

  11. 11.

    “I” for incomparable.

  12. 12.

    As is explained in [Ful80] and [Del80], Zariski gave a proof of Theorem 4.20, but it was incomplete because it relied on a flawed argument of Severi.

  13. 13.

    An alternative proof of Theorem 4.21 can be gotten by using (4.3.6), followed by a Segre embedding, to embed \({\mathcal{M}}_{0,n}\) into a large projective space. Then one projects onto a suitably chosen hyperplane. The composite is the desired embedding. We leave the details to the reader.

  14. 14.

    Indeed, in a sense which can be made precise, “most” finite index subgroups of SL(2, Z) are not congruence subgroups. (See [LS03] for a discussion of this topic.)

  15. 15.

    In particular, this argument shows, in a roundabout way, that as an abstract group Γfree(4) is isomorphic to the free group on two generators.

  16. 16.

    The definition of semistable was given in §2, in the discussion preceding Theorem 2.8.

  17. 17.

    Recall that a number field K is totally real if the images of all its complex embeddings are contained in the real numbers.

  18. 18.

    Recall that a CM field is a totally imaginary quadratic extension of a totally real field, the prototypical example of CM fields being imaginary quadratic fields.

References

  1. S. Abhyankar. Coverings of algebraic curves. Amer. J. Math., 79:825–856, 1957.

    Article  MathSciNet  MATH  Google Scholar 

  2. M. Artebani and G.-P. Pirola. Algebraic functions with even monodromy. Proc. Amer. Math. Soc., 133:(2)331–341, 2004.

    Google Scholar 

  3. C. Breuil, B. Conrad, F. Diamond, and R. Taylor. On the modularity of elliptic curves over Q: Wild 3-adic exercises. JAMS, 14:843–949, 2001.

    MathSciNet  MATH  Google Scholar 

  4. Beckmann. Ramified primes in the field of moduli of branched coverings of curves. J. Algebra, 125:236–255, 1989.

    Google Scholar 

  5. G. V. Belyi. On Galois extensions of a maximal cyclotomic field. Math. USSR Izvestija, 14:247–256, 1980.

    Article  MathSciNet  MATH  Google Scholar 

  6. G. V. Belyi. A new proof of the three point theorem. Sb. Math., 193(3-4):329–332, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  7. K. Behrend and B. Noohi. Uniformization of Deligne-Mumford curves. J. Reine Angew. Math., 599:111–153, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  8. V. Braungardt. Covers of moduli surfaces. Compositio Math., 140(4):1033–1036, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  9. P. Dèbes and J.-C. Douai. Algebraic covers: Field of moduli versus field of definition. Ann. Sci. ENS, 30(3):303–338, 1997.

    MATH  Google Scholar 

  10. P. Deligne. Le groupe fondamental du complément d’une courbe plane n’ayant que des points doubles ordinaires est abélien. In Lecture Notes in Math., Vol. 842,, pages 1–10. Séminaire Bourbaki, 1979-1980.

    Google Scholar 

  11. P. Deligne and D. Mumford. The irreducibility of the space of curves of given genus. Publ. Math. IHES, 36:75–109, 1969.

    MathSciNet  MATH  Google Scholar 

  12. P. Deligne and M. Rapoport. Les schémas de modules des courbes elliptiques. In Modular Functions of one variable II, Vol. 349, Springer Lecture Notes in Math., pages 143–316. Springer-Verlag, 1973.

    Google Scholar 

  13. W. Fulton. Hurwitz schemes and the irreducibility of moduli of algebraic curves. Ann. Math., 90:542–575, 1969.

    Article  MathSciNet  MATH  Google Scholar 

  14. W. Fulton. On the fundamental group of the complement of a node curve. Ann. Math. 111:407–409, 1980.

    Article  MathSciNet  MATH  Google Scholar 

  15. A. Grothendieck. Revêtements étales et groupe fondamental (SGA1), Vol. 224, Lect. Notes in Math., Springer-Verlag, 1971.

    Google Scholar 

  16. J. Harris and D. Mumford, On the Kodaira dimension of the moduli space of curves. Invent. Math., 67:23–86, 1982.

    Article  MathSciNet  MATH  Google Scholar 

  17. N. Katz. Travaux de Laumon, Séminaire Bourbaki, (691):105–132, 1987–1988.

    Google Scholar 

  18. C. Khare. Belyi parametrizations of elliptic curves and congruence defects. CRM Proc. Lecture Notes, 36:189–195, 2004.

    MathSciNet  Google Scholar 

  19. N. Katz and B. Mazur. Arithmetic moduli of elliptic curves, Vol. 108, Annals of Math. Studies, Princeton Univ. Press, 1985.

    Google Scholar 

  20. A. Lubotsky and D. Segal. Subgroup growth, Vol. 212, Progress in Math., Birkhauser-Verlag, 2003.

    Book  Google Scholar 

  21. R. C. Mason. Diophantine equations over function fields, Vol. 96, London Math. Soc. Lecture Notes Series. Cambridge University Press, 1984.

    MATH  Google Scholar 

  22. B. Mazur. Number theory as gadfly. Amer. Math. Monthly, 98(7):593–610, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  23. J. Oesterlé. Nouvelles approches du “théorème” de Fermat. In Asterisque, Vol. 161–162, pages 165–186. Séminaire Bourbaki, 1988/9.

    Google Scholar 

  24. K. Paranjape. A geometric characterization of arithmetic varieties. Proc. Indian. Acad. Sci. (Math), 112:1–9, 2002.

    Google Scholar 

  25. M. Raynaud. Revêtements de la drote affine en caractéristique p > 0 et conjecture de Abhyankar. Invent. Math., 116:425–462, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  26. M. Saidi. Revêtements modérés et groupe fondamental de graphes de groupes. Compositio Math., 107:319–338, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  27. S. Schroer. Curves with only triple ramification. Ann. Inst. Fourier, 53(7):2225–2241, 2003.

    Article  MathSciNet  Google Scholar 

  28. J.-P. Serre. Revêtements de courbes algébriques. Astérisque, Vol. 206, pages 167–182 Séminaire Bourbaki, 1992.

    Google Scholar 

  29. W. Stothers. Polynomial identities and Hauptmoduln. Quart. J. Math. Oxford (2), 32:349–370, 1981.

    Google Scholar 

  30. R. Taylor. l-adic representations associated to modular forms over imaginary quadratic fields. II. Invent. Math., 116:619–643, 1994.

    Google Scholar 

  31. R. Taylor. Galois representations. In Proc. ICM 2002, Vol. I, pages 449–474. Higher Ed. Press, Beijing, 2002.

    Google Scholar 

  32. R. Taylor. Galois representations. Ann. Math. Sci. Toulouse, 13(1):75–119, 2004.

    Google Scholar 

  33. R. Taylor and A. Wiles. Ring-theoretic properties of certain Hecke algebras. Ann. Math. (2), 141(3):553–572, 1995.

    Google Scholar 

  34. A. Weil. The field of definition of a variety. Amer. J. Math., 78:509–524, 1956.

    Article  MathSciNet  MATH  Google Scholar 

  35. A. Wiles. Modular elliptic curves and Fermat’s Last Theorem. Ann. Math. (2), 141(3):443–551, 1995.

    Google Scholar 

  36. L. Zapponi. A lower bound for the Belyi degree, Private oral communication, December 2008.

    Google Scholar 

  37. L. Zapponi. On the 1-pointed curves arising as étale covers of the affine line in positive characteristic. Math Zeit., 258(4):711–727, 2008.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA

    Wushi Goldring

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  1. Wushi Goldring
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Correspondence to Wushi Goldring .

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Editors and Affiliations

  1. , Department of Mathematics, Columbia University, Broadway 2990, New York, 10027, New York, USA

    Dorian Goldfeld

  2. , Department of Mathematics, City College of New York, New York, 10031, New York, USA

    Jay Jorgenson

  3. , Department of Mathematics, Yale University, Hillhouse Ave. 10, New Haven, 06520, Connecticut, USA

    Peter Jones

  4. , Department of Mathematics, California Institute of Technology, E. California Blvd. 1200, Pasadena, 91125, California, USA

    Dinakar Ramakrishnan

  5. , Department of Mathematics, University of California at Berkeley, Evans Hall 925, Berkeley, 94720-3840, California, USA

    Kenneth Ribet

  6. , Department of Mathematics, Harvard University, Science Center/One Oxford Street, Cambridge, 02138, Massachusetts, USA

    John Tate

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Goldring, W. (2012). Unifying themes suggested by Belyi’s Theorem. In: Goldfeld, D., Jorgenson, J., Jones, P., Ramakrishnan, D., Ribet, K., Tate, J. (eds) Number Theory, Analysis and Geometry. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-1260-1_10

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