Riemann Surface

A Riemann surface is a surface-like configuration that covers the complex plane with several, and in general infinitely many, "sheets." These sheets can have very complicated structures and interconnections (Knopp 1996, pp. 98-99). Riemann surfaces are one way of representing multiple-valued functions; another is branch cuts. The above plot shows Riemann surfaces for solutions of the equation

with , 3, 4, and 5, where  is the Lambert W-function (M. Trott).

The Riemann surface  of the function field  is the set of nontrivial discrete valuations on . Here, the set  corresponds to the ideals of the ring  of integers of  over . ( consists of the elements of  that are roots of monic polynomials over .) Riemann surfaces provide a geometric visualization of functions elements and their analytic continuations.

Schwarz proved at the end of nineteenth century that the automorphism group of a compact Riemann surface of genus  is finite, and Hurwitz (1893) subsequently showed that its order is at most  (Arbarello et al. 1985, pp. 45-47; Karcher and Weber 1999, p. 9). This bound is attained for infinitely many , with the smallest  of such an extremal surface being 3 (corresponding to the Klein quartic). However, it is also known that there are infinitely many genera for which the bound  is not attained (Belolipetsky 1997, Belolipetsky and Jones).

REFERENCES:

Arbarello, E.; Cornalba, M.; Griffiths, P. A.; and Harris, J. Geometry of Algebraic Curves, I. New York: Springer-Verlag, 1985.

Belolipetsky, M. "On the Number of Automorphisms of a Nonarithmetic Riemann Surface." Siberian Math. J. 38, 860-867, 1997.

Belolipetsky, M. and Jones, G. "A Bound for the Number of Automorphisms of an Arithmetic Riemann Surface." Math. Proc. Camb. Phil. Soc. 138, 289-299, 2005.

Borwein, J. M. and Corless, R. M. "Emerging Tools for Experimental Mathematics." Amer. Math. Monthly 106, 899-909, 1999.

Corless, R. M. and Jeffrey, D. J. "Graphing Elementary Riemann Surfaces." ACM Sigsam Bulletin: Commun. Comput. Algebra 32, 11-17, 1998.

Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Penguin, pp. 209-210, 2004.

Fischer, G. (Ed.). Plates 123-126 in Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig, Germany: Vieweg, pp. 120-123, 1986.

Hurwitz, A. "Über algebraische Gebilde mit eindeutigen Transformationen in sich." Math. Ann. 41, 403-442, 1893.

Karcher, H. and Weber, M. "The Geometry of Klein's Riemann Surface." In The Eightfold Way: The Beauty of the Klein Quartic (Ed. S. Levy). New York: Cambridge University Press, pp. 9-49, 1999.

Knopp, K. Theory of Functions Parts I and II, Two Volumes Bound as One, Part II. New York: Dover, pp. 99-118, 1996.

Krantz, S. G. "The Idea of a Riemann Surface." §10.4 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 135-139, 1999.

Kulkarni, R. S. "Pseudofree Actions and Hurwitz's  Theorem." Math. Ann. 261, 209-226, 1982.

Lehner, J. and Newman, M. "On Riemann Surfaces with Maximal Automorphism Groups." Glasgow Math. J. 8, 102-112, 1967.

Macbeath, A. M. "On a Curve of Genus 7." Proc. Amer. Math. Soc. 15, 527-542, 1965.

Mathews, J. H. and Howell, R. W. Complex Analysis for Mathematics and Engineering, 4th ed. Boston, MA: Jones and Bartlett, 2000.

Monna, A. F. Dirichlet's Principle: A Mathematical Comedy of Errors and Its Influence on the Development of Analysis. Utrecht, Netherlands: Osothoek, Scheltema, and Holkema, 1975.

Springer, G. Introduction to Riemann Surfaces, 2nd ed. New York: Chelsea, 1981.

Trott, M. "Visualization of Riemann Surfaces of Algebraic Functions." Mathematica J. 6, 15-36, 1997.

Trott, M. "Visualization of Riemann Surfaces IIa." Mathematica J. 7, 465-496, 2000.

 Trott, M. "Visualization of Riemann Surfaces." http://library.wolfram.com/examples/riemannsurface/.