Why were Abelian functions so important in the 19th century? Felix Klein, when discussing how the popularity of areas in mathematics rises and falls, mentions that in his youth Abelian functions were at the summit of mathematics, and that later on their popularity plummeted. I could hardly find anything on the net on Abelian functions and Wikipedia thinks that they are barely worth mentioning. What did the 19th century theory of these functions consist of, and why was it so important?
 A: Recall C. L. Siegel's rant, about the modern theory of abelian functions not having any functions in it.
From a point of view that would have made sense to Weierstrass, mathematics has "addition theorems", such as one first meets for sin and cos. One achievement of the 19th century was to classify these addition theorems, in "several variables" (which were of course complex variables for those guys), which were "algebraic". Scare quotes not too serious: this is one origin of today's theory of algebraic groups. To a first approximation, the theory of projective algebraic groups is the theory of abelian functions. Riemann had in fact written down enough theta functions so that abelian functions could be expressed in terms of their quotients. This was all a big and much-sought general framework extending the elliptic functions. The reason to do that was that, for example, indefinite integrals of square roots of cubics and quartics should have their excellent theory extended to quintics and beyond.
It turns out that abelian functions in at least two variables is not quite the formula-fest that the theory of elliptic functions is. Not for want of trying. We have the conceptual and geometric framework in the theory of abelian varieties. There are distinguished mathematicians who will tell you not to write down explicit functions. There is plenty of work, though, for example stimulated by soliton theory, explicit computation on curves of small genus, and so on.
A: For a really detailed answer to your question, see The Legacy of Niels Henrik
Abel, edited by O.A. Laudal and R. Piene (Springer 2004). In particular, there is a long introductory article by Christian Houzel, most of which can be viewed here.
Addendum. The complete article "The Work of Niels Henrik Abel" by
Christian Houzel may be found here.
