On a conjecture of Whittaker concerning uniformization of hyperelliptic curves

Authors:
Ernesto Girondo and Gabino González-Diez

Journal:
Trans. Amer. Math. Soc. **356** (2004), 691-702

MSC (2000):
Primary 30F10; Secondary 14H15

DOI:
https://doi.org/10.1090/S0002-9947-03-03441-X

Published electronically:
September 22, 2003

MathSciNet review:
2022716

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This article concerns an old conjecture due to E. T. Whittaker, aiming to describe the group uniformizing an arbitrary hyperelliptic Riemann surface as an index two subgroup of the monodromy group of an explicit second order linear differential equation with singularities at the values .

Whittaker and collaborators in the thirties, and R. Rankin some twenty years later, were able to prove the conjecture for several families of hyperelliptic surfaces, characterized by the fact that they admit a large group of symmetries. However, general results of the analytic theory of moduli of Riemann surfaces, developed later, imply that Whittaker's conjecture cannot be true in its full generality.

Recently, numerical computations have shown that Whittaker's prediction is incorrect for random surfaces, and in fact it has been conjectured that it only holds for the known cases of surfaces with a large group of automorphisms.

The main goal of this paper is to prove that having many automorphisms is not a necessary condition for a surface to satisfy Whittaker's conjecture.

**1.**Rolf Brandt and Henning Stichtenoth,*Die Automorphismengruppen hyperelliptischer Kurven*, Manuscripta Math.**55**(1986), no. 1, 83–92 (German, with English summary). MR**828412**, https://doi.org/10.1007/BF01168614**2.**Robert Brooks, Hershel M. Farkas, and Irwin Kra,*Number theory, theta identities, and modular curves*, Extremal Riemann surfaces (San Francisco, CA, 1995) Contemp. Math., vol. 201, Amer. Math. Soc., Providence, RI, 1997, pp. 125–154. MR**1429197**, https://doi.org/10.1090/conm/201/02606**3.**E. Bujalance, J. M. Gamboa, and G. Gromadzki,*The full automorphism groups of hyperelliptic Riemann surfaces*, Manuscripta Math.**79**(1993), no. 3-4, 267–282. MR**1223022**, https://doi.org/10.1007/BF02568345**4.**David V. Chudnovsky and Gregory V. Chudnovsky,*Computer algebra in the service of mathematical physics and number theory*, Computers in mathematics (Stanford, CA, 1986) Lecture Notes in Pure and Appl. Math., vol. 125, Dekker, New York, 1990, pp. 109–232. MR**1068536****5.**D.P. Dalzell,*A note on automorphic functions.*J. London Math. Soc.**5**(1930), 280-282.**6.**S.C. Dhar,*On the uniformization of a special kind of algebraic curves of any genus.*J. London Math. Soc.**10**(1935), 259-263.**7.**Brent Everitt,*A family of conformally asymmetric Riemann surfaces*, Glasgow Math. J.**39**(1997), no. 2, 221–225. MR**1460637**, https://doi.org/10.1017/S0017089500032109**8.**H. M. Farkas and I. Kra,*Riemann surfaces*, 2nd ed., Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York, 1992. MR**1139765****9.**L.R. Ford,*Automorphic functions.*Chelsea Publishing Company (1951).**10.**Y. Fuertes, G. González-Diez,*Smooth hyperelliptic double coverings of hyperelliptic surfaces.*Preprint.**11.**Ernesto Girondo and Gabino González-Diez,*On extremal Riemann surfaces and their uniformizing Fuchsian groups*, Glasg. Math. J.**44**(2002), no. 1, 149–157. MR**1892291**, https://doi.org/10.1017/S0017089502010108**12.**Joachim A. Hempel,*On the uniformization of the 𝑛-punctured sphere*, Bull. London Math. Soc.**20**(1988), no. 2, 97–115. MR**924235**, https://doi.org/10.1112/blms/20.2.97**13.**J. Hodgkinson,*Note on the uniformization of hyperelliptic curves.*J. London Math. Soc.**11**(1936), 185-192.**14.**Irwin Kra,*Accessory parameters for punctured spheres*, Trans. Amer. Math. Soc.**313**(1989), no. 2, 589–617. MR**958896**, https://doi.org/10.1090/S0002-9947-1989-0958896-0**15.**M. Mursi,*On the uniformisation of algebraic curves of genus .*Proc. Edinburgh Math. Soc.**2**(1930), 101-107.**16.**R.A. Rankin,*The differential equations associated with the uniformization of certain algebraic curves.*Proc. Roy. Soc. Edinburgh Sect. A**65**(1958), 35-62. MR**19:1172d****17.**Peter Turbek,*An explicit family of curves with trivial automorphism groups*, Proc. Amer. Math. Soc.**122**(1994), no. 3, 657–664. MR**1242107**, https://doi.org/10.1090/S0002-9939-1994-1242107-2**18.**E.T. Whittaker,*On the connexion of algebraic functions with automorphic functions.*Phil. Trans. Roy. Soc. London**192A**(1899), 1-32.**19.**E.T. Whittaker,*On hyperlemniscate functions. A family of automorphic functions.*J. London Math. Soc.**4**(1929), 274-278.**20.**J.M. Whittaker,*The uniformisation of algebraic curves.*J. London Math. Soc.**5**(1930), 150-154.**21.**E. T. Whittaker and G. N. Watson,*A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions: with an account of the principal transcendental functions*, Fourth edition. Reprinted, Cambridge University Press, New York, 1962. MR**0178117**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
30F10,
14H15

Retrieve articles in all journals with MSC (2000): 30F10, 14H15

Additional Information

**Ernesto Girondo**

Affiliation:
Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid 28049, Spain

Email:
ernesto.girondo@uam.es

**Gabino González-Diez**

Affiliation:
Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid 28049, Spain

Email:
gabino.gonzalez@uam.es

DOI:
https://doi.org/10.1090/S0002-9947-03-03441-X

Keywords:
Accessory parameters,
Schwarzian derivative,
uniformization of Riemann surfaces,
hyperelliptic curves

Received by editor(s):
July 23, 2002

Published electronically:
September 22, 2003

Additional Notes:
Both authors were supported in part by Grant BFM2000-0031, DGI.MCYT

Article copyright:
© Copyright 2003
American Mathematical Society