582 XIII Connections with Number Theory
was already studied by Heine (1847). There is indeed a whole world ofq-analysis,
which may be regarded as having the same relation to classical analysis as quan-
tum mechanics has to classical mechanics. (The choice of the letter ‘q’ nearly a
century before the advent of quantum mechanics showed remarkable foresight.) There
are introductions to this world in Andrewset al.[4] and Vilenkin and Klimyk [58].
For Macdonald’s conjectures concerningq-analogues of orthogonal polynomials, see
Kirillov [36].
Althoughq-analysis always had its devotees, it remained outside the mainstream
of mathematics until recently. Nowit arises naturally in the study ofquantum groups,
which are not groups butq-deformations of the universal enveloping algebra of a Lie
algebra.
The Rogers–Ramanujan identities were discovered independently by Rogers
(1894), Ramanujan (1913) and Schur (1917). Their romantic history is retold in
Andrews [2], which contains also generalizations. For the applications of the iden-
tities in statistical mechanics, see Baxter’s article (pp. 69–84) in Andrewset al.[3].
(The same volume contains other interesting articles on mathematical developments
arising from Ramanujan’s work.)
The Jacobi triple product formula was derived in Chapter XII as the limit of a
formula for polynomials. Andrews [1] has given a similar derivation of the Rogers–
Ramanujan identities. This approach has foundapplications and generalizations in
conformal field theory, with the two sides of the polynomial identity corresponding
to fermionic and bosonic bases for Fock space; see Berkovich and McCoy [9].
These connections go much further than the Rogers–Ramanujan identities. There
is now a vast interacting area which involves, besides the theory of partitions, solv-
able models of statistical mechanics, conformal field theory, integrable systems in
classical and quantum mechanics, infinite-dimensional Lie algebras, quantum groups,
knot theory and operator algebras. For introductory accounts, see [45], [10] and
various articles in [24] and [27]. More detailed treatments of particular aspects are
given in Baxter [8], Faddeev and Takhtajan[26], Jantzen [33], Jones [34], Kac [35]
and Korepinet al.[38].
For the Hardy–Ramanujan–Rademacher expansion forp(n), see Rademacher [46]
and Andrews [2]. An interesting proof by means of probability theory for the first term
of the expansion has been given by B ́aez-Duarte [5].
The definition of birational equivalence in§3 is adequate for our purposes, but has
been superseded by a more general definition in the language of ‘schemes’, which is
applicable to algebraic varieties of arbitrary dimension without any given embedding
in a projective space. For the evolution of the modern concept, seeCi ̆ ̆zm ́ar [18].
The history of the discovery of the group law on a cubic curve is described by
Schappacher [48].
Several good accounts of the arithmetic of elliptic curves are now available; e.g.,
Knapp [37] and the trilogy [52], [50], [51]. Although the subject has been transformed
in the past 25 years, the survey articles by Cassels [16], Tate [55] and Gelbart [28] are
still of use. Tate gives a helpful introduction, Cassels has many references to the older
literature, and Gelbart explains the connection with the Langlands program, for which
see also Gelbart [29].