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).

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

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Weisstein, Eric W. "Riemann Surface." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/RiemannSurface.html