8 Further Remarks 219M=
(
AB
CD
)
issymplectic,i.e.ifMtJM=J,where
J=
(
OI
−IO
)
,
then the linear fractional transformationZ→(AZ+B)(CZ+D)−^1 ,mapsHnonto
itself. Siegel’s modular groupΓnis the group of all such transformations. The gener-
alized upper half-planeHnis itself just a special case of the vast theory of symmetric
Riemannian spaces initiated by E. Cartan (1926/7). See Siegel [54] and Helgason [25].
The development of non-Euclidean geometry is traced in Bonola [10]. (This
edition also contains translations of works by Bolyai and Lobachevski.) The basic
properties of Poincar ́e’s model, here only stated, are proved in the books of Katok [29]
and Beardon [7].
For the connection between continued fractions and geodesics, see Artin [5] and
Sheingorn [53]. For the Markov spectrum see not only the books of Cassels [13] and
Rockett and Szusz [45], but also Cusick and Flahive [16] and Baragar [6].
The theory of continued fractions for formal Laurent series is developed further in
de Mathan [37]. The corresponding theory of Diophantine approximation is surveyed
in Lasjaunias [35]. The polynomial Pell equation is discussed by Schmidt [48]. For for-
mal Laurent series there is a multidimensional generalization which is quite different
from those for real numbers; see Antoulas [4].
Roth’s theorem and Schmidt’s subspace theorem are proved in Schmidt [47]. See
also Faltings and W ̈ustholz [20] and Evertse [19]. Nevanlinna’s theory of the value dis-
tribution of meromorphic functions is treated in the recent book of Cherry and Ye [14].
For Vojta’s work see, for example, [59] and [60]. It should be noted, though, that
this area is still in a state of flux, besides using techniques beyond our scope. For an
overview, see Lang [34].
Siegel’s theorem on Diophantine equations with infinitely many solutions is proved
with the aid of non-standard analysis by Robinson and Roquette [44]; the proof is re-
produced in Stepanov [57]. The theorem is discussed from the standpoint ofDiophan-
tine geometryin Serre [50]. Any algebraic curve overQof genus zero which has a
nonsingular rational point can be parametrized by rational functionseffectively;see
Poulakis [43].
It is worth noting that ifF(x,y)is a polynomial with rational coefficients which
is irreducible overQ, but not overC, then the curveF(x,y)=0 has at most finitely
many rational points. For any rational point is a common root of at least two distinct
complex-irreducible factors ofFand any two such factors have at most finitely many
common complex roots.
In conclusion we mention some further applications of continued fractions. A pro-
cedure, due to Vincent (1836), for separating the roots of a polynomial with integer
coefficients has acquired some practicalvalue with the advent of modern computers.
See Alesina and Galuzzi [3].
Continued fractions play a role in the small divisor problems of classical mechan-
ics. As an example, suppose the functionfis holomorphic in some neighbourhood