Automorphic function

A meromorphic function of several complex variables that is invariant under some discrete group of transformations $ \Gamma $ of analytic transformations of a given complex manifold $ M $ :

$$ f ( \gamma ( x ) ) = f ( x ) , x \in M , \gamma \in \Gamma . $$

Automorphic functions are often defined so as to include only functions defined on a bounded connected domain $ D $ of the $ n $-dimensional complex space $ \mathbf C ^{n} $ that are invariant under a discrete group $ \Gamma $ of automorphisms of this domain.

The quotient space $ X = M / \Gamma $ can be given a complex structure and automorphic functions are then meromorphic functions on $ X $ . A large number of cases studied concern a space $ X $ with a compactification $ \bar X $ . It is then natural to include in the definition of an automorphic function the requirement that it can be extended to the entire space $ \bar X $ as a meromorphic function. If $ M = D $ (i.e. $ M $ is a bounded connected domain), this condition must be required for $ n = 1 $ only (if $ n > 1 $ or if $ M / \Gamma $ is compact, the condition is automatically fulfilled). It can readily be shown that the automorphic functions constitute a field $ K ( \Gamma ) $ and the study of this field is one of the main tasks in the theory of automorphic functions.

Automorphic functions of a single variable have been very thoroughly studied. The theoretical foundations were laid by F. Klein [1] and H. Poincaré [2] in the 19th century. The manifold $ M $ usually considered at that time is a simply-connected domain. Three cases are distinguished: $ M = P ^{1} ( \mathbf C ) $ (the complex projective line, or the Riemann sphere), $ M = \mathbf C $ and $ M = H $ (the upper half-plane $ \{ {z \in \mathbf C} : { \mathop{\rm Im}\nolimits z > 0} \} $ ). In the first case the discrete groups $ \Gamma $ are finite, the curves $ M / \Gamma $ are algebraic curves of genus 0 (cf. Genus of a curve) and, consequently, the automorphic functions generate a field of rational functions. Examples of automorphic functions in the case $ M = \mathbf C $ are periodic functions (thus, the function $ e ^ {2 \pi i z} $ is invariant under the translation group $ \{ {z \rightarrow z + n} : {n \in \mathbf Z} \} $ ) and, in particular, elliptic functions. In this latter case, the curve $ M / \Gamma $ is compact and is an elliptic curve, while the field $ K( \Gamma ) $ is the field of all algebraic functions on $ M / \Gamma $ . Finally, for $ M = H $ and a discrete group $ \Gamma $ such that $ M / \Gamma $ is compact or has a finite volume (in the Poincaré metric), $ M / \Gamma $ is an algebraic curve and $ K ( \Gamma ) $ is again the field of algebraic functions on $ M / \Gamma $ . The genus $ g $ of this curve may be determined by constructing a fundamental domain for $ \Gamma $ in the form of a polygon in the upper half-plane $ H $ (here regarded as the Lobachevskii plane). The basic method for constructing an automorphic function in this situation is to consider the quotient of two automorphic forms (cf. Automorphic form) of the same, sufficiently large, weight. The method is due to Poincaré, who used it to prove the results mentioned above concerning the structure of the fields of automorphic functions [2], [3], [4]. An analogous construction for elliptic functions is to represent such functions in terms of quotients of theta-functions. It can be shown, using uniformization theory, that all fields of algebraic functions of a single variable are obtained in this way [3].

These results, which were obtained as early as the 19th century, give a full description of the fields of automorphic functions for $ n = 1 $ and of the groups $ \Gamma $ such that the space $ H / \Gamma $ has finite volume. The case of groups $ \Gamma $ for which $ H / \Gamma $ has infinite volume (Kleinian groups) is much more difficult; the problems involved are still being intensively investigated [5], [6].

In the 20th century the theory of automorphic functions concentrates on functions of several variables. Perhaps the only case of automorphic functions of $ n $ variables studied in detail in the 19th century concerned Abelian functions, which are related to Abelian varieties in a way similar to the relationship between elliptic functions and elliptic curves [1], [7]. The first example of automorphic functions of $ n $ variables on a bounded domain $ D $ are the modular functions of C.L. Siegel [7] (cf. Modular group). Their domain of definition is an $ n $-dimensional generalization of the upper half-plane $ H $ , and is one of the main examples of a bounded, symmetric domain. Siegel must also be credited with the first general results obtained concerning arbitrary automorphic functions on a bounded domain $ D $ . He generalized Poincaré's construction of automorphic functions, mentioned above, and showed that the field $ K ( \Gamma ) $ always contains at least $ n $ algebraically independent functions.

Subsequent efforts were aimed at exhibiting domains $ D $ and groups $ \Gamma $ for which the following theorem on algebraic relations is true. If $ f _{1}, \dots, f _{n} $ are algebraically independent automorphic functions, then the field $ K ( \Gamma ) $ is a finite algebraic extension of the field of rational functions $ \mathbf C ( f _{1}, \dots, f _{n} ) \subset K ( \Gamma ) $ .

At the time of writing (1977) this theorem was proved for the following cases: 1) if the quotient space $ D / \Gamma $ is compact [7]; 2) if the group $ \Gamma $ is pseudo-concave [8]; and 3) if $ D $ is a symmetric domain and $ \Gamma $ is an arithmetic group. A pseudo-concave group is defined as follows. Let $ X $ be a subdomain of a domain $ D $ with closure also contained in $ D $ . In this situation a boundary point $ x _{0} \in \partial X $ is said to be pseudo-concave if for any open neighbourhood $ U $ of $ x _{0} $ and for any function $ \phi (x) $ regular in $ U $ there exists a point $ x \in U \cap X $ for which $ | \phi (x) | \geq | \phi ( x _{0} ) | $ . A group $ \Gamma $ is said to be pseudo-concave if there exists a subdomain $ X \subset D $ such that each boundary point $ x \in \partial X $ can be transformed by means of an element of $ \Gamma $ into an interior point of $ X $ or into a pseudo-concave point of the boundary $ \partial X $ .

The nature and the properties of the algebraic varieties occurring in the theory of automorphic functions of $ n $ variables have not been intensively studied, as distinct from the case of a single variable.

Important generalizations of the concept of automorphic functions — automorphic forms, theta-functions (cf. Theta-function) and certain other generalizations — are all special cases of the following general construction. Consider a fibre bundle (cf. Fibration) $ L $ over $ M $ and an action of a group $ \Gamma $ on it. It is then possible to consider the sections of $ L $ that are invariant under $ \Gamma $ . An automorphic function is obtained if the fibre bundle $ L $ and the action of the group $ \Gamma $ are both trivial.

The study of automorphic functions revealed the important role played by the group of automorphisms of a domain $ D $ . It is in this way that the concepts and methods of the theory of automorphic functions were applied in the theory of algebraic groups, in which they play an important part in the description of infinite-dimensional representations [10].

From the very beginning of its development, the theory of automorphic functions has been connected in numerous ways with other branches of mathematics. This applies in particular to algebraic geometry. In addition to the results discussed above, methods in the theory of automorphic functions are important in the study of moduli varieties for objects such as algebraic curves and Abelian varieties. Automorphic functions are also of importance in number theory. At the time of writing they are the only tool in the study of zeta-functions of algebraic varieties [11]. Another very promising number-theoretical direction in the theory of automorphic functions is the study of $ p $-adic automorphic functions and forms [9]. Finally, one must mention the application of automorphic functions to the study of ordinary differential equations in a complex domain [12] and in the construction of solutions of algebraic equations of degrees higher than four.

References

[1]	F. Klein, "Development of mathematics in the 19th century" , 1 , Math. Sci. Press (1979) pp. Chapt.8 (Translated from German) MR0529278 MR0549187 Zbl 0411.01009
[2]	H. Poincaré, , Oeuvres de H. Poincaré , 4 , Gauthier-Villars (1916–1965)
[3]	L.R. Ford, "Automorphic functions" , Chelsea, reprint (1951) MR1522111 Zbl 55.0810.04 Zbl 46.0621.01 Zbl 45.0693.07
[4]	H. Bateman (ed.) A. Erdélyi (ed.) , Higher transcendental functions , 3. Automorphic functions , McGraw-Hill (1955) MR0066496 Zbl 0157.11901 Zbl 0143.29202 Zbl 0146.09301
[5]	J. Hadamard, "La géometrie non-euclidienne dans la théorie des fonctions automorphes" , Moscow (1952) (In Russian; translated from French)
[6]	I. Kra, "Automorphic forms and Kleinian groups" , Benjamin (1972) MR0357775 Zbl 0253.30015
[7]	C.L. Siegel, "Automorphe Funktionen in mehrerer Variablen" , Math. Inst. Göttingen (1955)
[8]	A. Andreotti, H. Grauert, "Algebraische Körper von automorphen Funktionen" Nachr. Akad. Wiss. Göttingen , 3 (1961) MR0132211 Zbl 0154.33604 Zbl 0096.28001
[9]	J.-P. Serre (ed.) P. Deligne (ed.) W. Kuyk (ed.) , Modular functions of one variable. 1–3 , Lect. notes in math. , 320; 349; 350 , Springer (1973) MR0323724
[10]	H. Jacquet, R.P. Langlands, "Automorphic forms on GL(2)" , Springer (1970–1972)
[11]	G. Shimura, "Introduction to the arithmetic theory of automorphic functions" , Math. Soc. Japan (1971) MR0314766 Zbl 0221.10029
[12]	V.V. Golubev, "Vorlesungen über Differentialgleichungen im Komplexen" , Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian) MR0100119

Comments

The result mentioned above that the field $ K ( \Gamma ) $ is a finite algebraic extension of a field of rational functions $ \mathbf C ( f _{1}, \dots, f _{n} ) $ (the theorem on algebraic relations) in the case of a symmetric domain $ D $ and an arithmetic group $ \Gamma $ is due to, independently, W.L. Baily jr. and A. Borel [a6], and I.I. Pyatetskii-Shapiro [a7].

Let $ X $ be some kind of space (e.g. complex- or real-analytic, a smooth manifold), $ \Gamma $ a group of automorphisms of $ X $ and $ H $ a group acting on a space $ V $ . Let $ \mathop{\rm Mor}\nolimits ( X , H ) $ be the set of morphisms from $ X $ into $ H $ . An automorphy factor of $ \Gamma $ is a $ 1 $-cocycle (crossed homomorphism) $ j $ of $ \Gamma $ with values in $ \mathop{\rm Mor}\nolimits ( X , H ) $ . That means it is a mapping $ j : X \times \Gamma \rightarrow H $ such that $ j ( x , \gamma \gamma ^ \prime ) = j ( x , \gamma ) j ( x \gamma , \gamma ^ \prime ) $ . An example is the Jacobian of $ j $ as a diffeomorphism $ X \rightarrow X $ (chain rule). An automorphic form of type $ j $ is now a morphism $ f : X \rightarrow V $ such that $ f (x) = j ( x , \gamma ) f ( x \gamma ) $ . Taking the Jacobian as an automorphy factor and $ H = \mathop{\rm GL}\nolimits ( \mathbf C ) $ acting on $ \mathbf C $ via the $ m $-th power of the determinant one recovers the more classical notion of an automorphic form of weight $ m $ , cf. Automorphic form. The automorphy factor $ j $ can be used to define an action of $ \Gamma $ on $ X \times V $ by $ ( x , v ) \gamma = ( x \gamma , j ( x , \gamma ) v ) $. If $ \Gamma $ now operates freely on $ X \times V $ as a properly discontinuous group of transformations, then $ ( X \times V ) / \Gamma $ is a fibre bundle over $ X / \Gamma $ with fibre $ V $ and the automorphic forms are the cross-sections of this bundle, or, equivalently, the $ \Gamma $-equivariant cross-sections of the trivial bundle $ X \times V \rightarrow X $ .

In a still more group-theoretical setting let $ G $ be a real semi-simple Lie group with Lie algebra $ \mathfrak g $ . Identify the universal enveloping algebra $ U \mathfrak g $ of $ \mathfrak g $ with the right-invariant differential operators $ D ( G ) $ on $ G $ by extending the mapping which assigns to $ a \in \mathfrak g $ the corresponding right-invariant vector field. Let $ K $ be a maximal compact subgroup of $ G $ and $ \Gamma $ a discrete subgroup and let $ \rho : K \rightarrow \mathop{\rm GL}\nolimits (V) $ be a representation of $ K $ . A smooth vector-valued function $ f : G \rightarrow V $ is called an automorphic form for $ \Gamma $ if $ f ( k g \gamma ) = \rho ( k ) f ( g ) $ , $ ( Z \mathfrak g ) f $ is a finite vector space, where $ Z ( \mathfrak g ) \subset U \mathfrak g = D ( G ) $ is the centre of $ U \mathfrak g $ , and $ f $ satisfies a certain growth condition. The link with the notion "automorphic form of type j" discussed just above is provided by $ j $ , the left coset space of $ X = K \smn G $ in $ K $ , and a canonical automorphy factor (with $ G $ ) which can be defined in this setting. Cf. [a1] for more details on all this.

Besides the applications of automorphic functions in ordinary differential equations and algebraic equations mentioned above there is also a most revealing connection between the harmonic analysis of functions automorphic with respect to a discrete subgroup of $ H = K _ {\mathbf C} $ and Lax–Philips scattering theory applied to the non-Euclidean wave equation, cf. [a4], [a5].

For more material closely related to automorphic forms and automorphic functions, cf. also the articles Modular form; Modular function; Fuchsian group; Discrete subgroup; Discrete group of transformations.

References

[a1]	A. Borel, "Introduction to automorphic forms" A. Borel (ed.) G.D. Mostow (ed.) , Algebraic groups and discontinuous subgroups , Proc. Symp. Pure Math. , 9 , Amer. Math. Soc. (1966) pp. 199–210 MR0207650 Zbl 0195.09501 Zbl 0191.09601
[a2]	R. Fricke, F. Klein, "Vorlesungen über die Theorie der automorphen Funktionen" , 1–2 , Teubner (1926) MR0183872 Zbl 32.0430.01 Zbl 43.0529.08 Zbl 42.0452.01
[a3]	A. Borel (ed.) W. Casselman (ed.) , Automorphic forms, representations and $ \mathop{\rm SL}\nolimits _{2} ( \mathbf R ) $ -functions , Proc. Symp. Pure Math. , 33:1–2 , Amer. Math. Soc. (1979) MR0546606 MR0546586
[a4]	L.D. Faddeev, B.S. Pavlov, "Scattering theory and automorphic functions" Proc. Steklov Inst. Math. , 27 (1972) pp. 161–198 MR0320781 Zbl 0343.35004
[a5]	P.D. Lax, R.S. Phillips, "Scattering theory for automorphic functions" Bull. Amer. Math. Soc. (New Ser.) , 2 (1980) pp. 261–296 MR0555264 Zbl 0442.10018
[a6]	W.L., jr. Baily, A. Borel, "Compactifications of arithmetic quotients of bounded symmetric domains" Ann. of Math. , 84 (1966) pp. 442–528 MR0216035
[a7]	I.I. Pyatetskii-Shapiro, "Arithmetic groups on complex domains" Russ. Math. Surveys , 19 (1964) pp. 83–109