A (Very Brief) Course of
Modern Analysis
In honor of today's anniversary of the 1873 birth of Edmund Taylor Whittaker, here are some references to a topic that still interests some mathematicians of today.
From A Course of Modern Analysis, by E. T. Whittaker and G. N. Watson, Fourth Edition, Cambridge University Press, 1927, reprinted 1969:
Section 20.7 "…the fact, that x and y can be expressed as one-valued functions of the variable z, makes this variable z of considerable importance… z is called the uniformizing variable of the equation…. When the genus of the algebraic curve f(x,y) = 0 is greater than unity, the uniformisation can be effected by means of what are known as automorphic functions. Two classes of such functions of genus greater than unity have been constructed, the first by Weber…(1886), the second by Whittaker…(1898)…."
The topic of uniformisation of algebraic curves has appeared frequently lately in connection with Wiles's attack on Fermat's Last Theorem. See, for instance, Lang's 1995 AMS Notices article
"Shimura's… insight was that the ordinary modular functions for a congruence subgroup of SL2(Z) suffice to uniformize elliptic curves defined over the rationals."
"The property of an elliptic curve [over Q] of being parameterized by modular functions is one way of defining a modular elliptic curve, and the Taniyama-Shimura conjecture asserts that every elliptic curve is modular."
For a deeper discussion of uniformisation in the context of Wiles's efforts, see "Elliptic curves and p-adic uniformisation," by H. Darmon, 1999.
For a more traditional approach to uniformisation, see "On the uniformisation of algebraic curves," by Yu. V. Brezhnev (24 May, 2002), which cites two of Whittaker's papers on automorphic functions (from 1898 and 1929) and a 1930 paper, "The uniformisation of algebraic curves," by J. M. Whittaker, apparently E. T. Whittaker's son.