Modular function

elliptic modular function, of one complex variable

An automorphic function of a complex variable , associated with the group of all fractional-linear transformations of the form

(1)

where are real integers (this group is called the modular group). The transformations of transform the real axis into itself and the domain of definition of a modular function can be regarded as being the upper half-plane . The group is generated by the two transformations , .

Figure: m064430a

A fundamental domain of the modular group is depicted in Fig. a; this is the curvilinear quadrangle with vertices , , , two sides of which, and , are segments of the lines and , respectively, and is an arc of the circle . and are included in , and are not. The images of under all possible mappings of cover the half-plane without intersections.

The study of modular functions began in the 19th century in connection with the study of elliptic functions and preceded the appearance of the general theory of automorphic functions. In the theory of modular functions the following theta-series are used as basic modular forms:

where and the asterisk means that the null pair is omitted. According to the terminology of K. Weierstrass these are relative invariants, playing a major role in his theory of elliptic functions (see Weierstrass elliptic functions), and is also called the discriminant. From the point of view of the theory of automorphic functions (cf. Automorphic function; Automorphic form) these are automorphic forms of weights , and , respectively, associated with the modular group. The fundamental modular form has the form

(2)

is also called the absolute invariant. It is regular in the upper half-plane and in the interior of the fundamental domain it takes each finite value, except and , precisely once; in addition, , .

The modular function plays a major role in the theory of elliptic functions, allowing one to determine the periods , with respect to given Weierstrass relative invariants , , , and, consequently, to construct all Weierstrass elliptic functions. If is the unique solution in the fundamental domain of the equation

then for , one has , ; for one has , and is determined by the equation

for one has , and is determined by the equation

For the construction of Jacobi elliptic functions, instead of it is more convenient to use

(3)

also called a modular function. By the same token, is an automorphic function only relative to the subgroup of , where consists of all transformations of the form (1) in which (as an extra condition) and are odd numbers and and are even. The fundamental domain of is depicted in Fig. b; this is the curvilinear quadrangle with vertices , , , , two sides of which, and , are segments of the lines and , respectively, and and are arcs of the circles and , respectively. The parts of the boundary to the left of the imaginary axis are included and and are not included.

Figure: m064430b

The function is also regular in the upper half-plane . In the interior of it takes each finite value, except and , precisely once; in addition, and . For the construction of a Jacobi elliptic function of given modulus the value , or , uniquely defined by the equation , is required. In practice, in the normal case , one first determines , where , and then constructs a solution of this equation in the form of a series . The modular functions and are related by

The modular function gives the most convenient representation of the conformal classes of Riemann surfaces of elliptic functions (cf. Riemann surfaces, conformal classes of), when the genus and the Euler characteristic . Corresponding to each there is a solution of , which determines a conformal class and the corresponding field of elliptic functions. For example, corresponds to a period parallelogram in the form of a rhombus with angles and , and corresponds to a square. Modular functions have also been applied in the study of conformal mapping; boundary properties of analytic functions and cluster sets (cf. Cluster set). The modular function gives a conformal mapping of the left half of the fundamental domain (Fig. a), that is, the curvilinear triangle , onto the upper half-plane , where , and are mapped to , and , respectively. The modular function conformally maps the curvilinear triangle (Fig. b) onto the upper half-plane, where , and are mapped to , and , respectively.

In geometric questions it is often more convenient to take the unit disc as the domain of the modular functions. The modular group (1) is then replaced by the modular group of automorphisms of the unit disc. For example, it is convenient to apply the fractional-linear transformation

which maps the upper half-plane onto the unit disc , where , and are mapped to , and , respectively, on the unit circle (Fig. c).

Figure: m064430c

Then the composite function is a modular function that is regular in the unit disc and takes there all values except , and . It conformally maps the curvilinear triangle (Fig. c) onto the upper half-plane . It is precisely this modular function that is used in the proof of the Picard theorem and in a number of geometric questions.

References

[1]	A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , 1 , Springer (1964) pp. Chapt.8
[2]	N.I. Akhiezer, "Elements of the theory of elliptic functions" , Amer. Math. Soc. (1990) (Translated from Russian)
[3]	L.R. Ford, "Automorphic functions" , Chelsea, reprint (1951)
[4]	F. Klein, R. Fricke, "Vorlesungen über die Theorie der elliptischen Modulfunktionen" , 1–2 , Teubner (1890–1892)

Comments

When considering the upper half-plane and the modular group acting on it, the point and the rational points on the real axis are often referred to as cusps.

More generally, consider the group of invertible complex -matrices, , and the corresponding fractional-linear transformations

(a1)

The fractional-linear transformation (a1) is called parabolic if it is and the associated matrix has two equal eigen values (cf. also Fractional-linear mapping). This is equivalent to saying that the Jordan canonical form is of the form , or, if is also imposed, that . Now let be some discrete subgroup of . A point is called a cusp of if there is a parabolic element of which has as a fixed point.

The cusps of are precisely the points of . To aid visualization, cf. Fig. a, the point is written as ( the point in Fig. a).

Let be the extended upper half-plane , where . The action of on naturally extends to , and all the points of form one orbit. The translates of the fundamental region of Fig. aform a tesselation of (or of ), called the modular tesselation. Each translate , , is called a modular triangle. In the special points , , , six modular triangles meet; in the special points , , , two modular triangles meet; and in a cusp (a point of ) countably infinite many modular triangles meet (at angle 0; whence the terminology "cusp" ).

The modular functions form a field. Indeed, this is the field , where is the fundamental modular function (2) above.

Let be a subgroup of finite index in . The quotient can be given a natural complex structure making it a compact Riemann surface, cf. e.g. [a1], Chapt. IV, § 6. This is a natural compactification of . For one finds the Riemann sphere (of genus zero). For the principal congruence subgroups

the resulting quotients , the modular curves , for have genus 0, 0, 0, 0, 1, 3, 5, 10, 13, 26, 25, respectively. For the general formula cf. Modular curve.

A modular function for a subgroup of finite index of is a complex meromorphic function on such that for , , and such that at a rational cusp , , admits an expansion of the form

for some , natural number and . This is valid for with large enough. This last condition reflects the requirement that also defines a meromorphic function on the compactification of , cf. Automorphic function. In case this last requirement takes the following form: There is an such that for , , has an expansion of the form

References

[a1]	B. Schoeneberg, "Elliptic modular functions" , Springer (1974)
[a2]	G. Shimura, "Introduction to the arithmetic theory of automorphic functions" , Princeton Univ. Press (1971)
[a3]	R.A. Rankin, "Modular forms and functions" , Cambridge Univ. Press (1977)
[a4]	S. Lang, "Elliptic functions" , Addison-Wesley (1973)