218 IV Continued Fractions and Their Uses
The continued fraction construction for the representation of a primep≡1mod4
as a sum of two squares is due to Legendre. Some other constructions are given in
Chapter V of Davenport [17] and in Wagon [61]. A construction for the representation
of any positive integer as a sum of four squares is given by Rousseau [46].
The modular group is the basic example of aFuchsian group, i.e. a discrete sub-
group of the groupPSL 2 (R)of all linear fractional transformationsz → (az+
b)/(cz+d),wherea,b,c,d ∈Randad−bc=1. Fuchsian groups are studied
from different points of view in the books of Katok [29], Beardon [7], Lehner [36],
and Vinberg and Shvartsman [58].
The significance of Fuchsian groups stems in part from the uniformization theo-
rem, which characterizes Riemann surfaces. ARiemann surfaceis a 1-dimensional
complex manifold. Two Riemann surfaces areconformally equivalentif there is a
bijective holomorphic map from one to the other. Theuniformization theorem,first
proved by Koebe and Poincar ́e independently in 1907, says that any Riemann surface
is conformally equivalent to exactly one of the following:
(i) the complex planeC,
(ii) the Riemann sphereC∪{∞},
(iii) the cylinderC/G,whereGis the cyclic group generated by the translation
z→z+1,
(iv) a torusC/G,whereGis the abelian group generated by the translationsz→z+ 1
andz→z+τfor someτ∈H (the upper half-plane),
(v) a quotient spaceH/G,whereGis a Fuchsian group which actsfreelyonH,
i.e. ifz∈H,g∈Gandg=I,theng(z)=z.
(It should be noted that, since the modular group does not act freely onH,thecor-
responding ‘Riemann surface’ isramified.) For more information on the uniformiza-
tion theorem, see Abikoff [1], Bers [9], Farkas and Kra [21], Jost [27], Beardon and
Stephenson [8], and He and Schramm [24].
For the equivalence between quadratic fields and binary quadratic forms, see
Zagier [63]. The class numberh(d)of the quadratic fieldQ(
√
d)has been deeply
investigated, originally by exploiting this equivalence. Dirichlet (1839) obtained an
analytic formula forh(d)with the aid of his theorem on primes in an arithmetic pro-
gression (which will be proved in Chapter X). A clearly motivated proof of Dirichlet’s
formula is given in Hasse [23], and there are some interesting observations on the
formula in Stark [56].
It was conjectured by Gauss (1801), in the language of quadratic forms, that
h(d)→∞asd→−∞. This was first proved by Heilbronn (1934). Siegel (1935)
showed that actually
logh(d)/log|d|→ 1 /2asd→−∞.Generalizations of these results to arbitrary algebraic number fields are given in books
on algebraic number theory, e.g. Narkiewicz [38].
Siegel (1943) has given a natural generalization of the modular group to higher
dimensions. Instead of the upper half-planeH, we consider the spaceHnof all com-
plexn×nmatricesZ=X+iY,whereX,Yare real symmetric matrices andYis
positive definite. If the real 2n× 2 nmatrix