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Lectures on Modular Forms
By Joseph Lehner Dover Publications, Inc.
Copyright © 2017 Joseph Lehner
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ISBN: 978-0-486-82140-5
CHAPTER 1
Modular Forms
1 The Modular Group and Some Subgroups
Throughout these lectures we shall use the letter Γ to denote the modular group, which is the group of linear-fractional transformations
τ' = aτ + b/cτ + d; a, b, c, d, ε Z, ad - bc = 1 (1)
Here Z is the set of rational integers. In group theory this group is referred to as LF (2, Z). Though the transformations (1) are the object of our study, it is much more convenient to use matrices. The group
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
is not quite isomorphic to Γ, but we have
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
We regard Γ as the matrix group SL (2, Z) in which each matrix is identified with its negative.
We shall be concerned with certain subgroups of Γ, all of finite index:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
and some others. Here it is necessary to be careful about the relation of the matrix to the transformation. Clearly, if -I is in the matrix group, (2) holds; otherwise the two are isomorphic. The first case occurs in the first 3 groups listed above, but -I]ITY [not member of] Γ'. In general, we shall make no distinction between the matrix and transformation groups, trusting to the context to keep things straight. For applications to number theory the group Γ0 (n) is especially important.
We shall denote by G an arbitrary subgroup of finite index in Γ. The group G is discrete, that is, it contains no infinite sequence of distinct matrices that converges to the identity matrix. Almost all of what we do goes over with minor changes to discrete groups of linear-fractional transformations with real coefficients provided they are finitely generated, but we shall not be concerned with the more general case except in Chapter IV.
The geometric theory of linear-fractional transformations is essential in our work. We shall deal only with linear-fractional transformations with real coefficients and determinant 1; call the group of such transformations Ω. An element of Ω maps the upper half-plane H on itself and the real axis on itself, and conversely any transformation with these properties can be written with real coefficients and determinant 1 and so belongs to Ω. We also have the classification into 3 types (disregarding the identity):
elliptic if |a + d| < 2
parabolic if |a + d| = 2
hyperbolic if |a + d| > 2. (3)
The interpretation as noneuclidean motions is familiar. (cf. ch. I notes, No. 1, p. 16). An elliptic transformation has two complex-conjugate fixed points, one lying in H; a hyperbolic transformation, two real (distinct) fixed points; a parabolic transformation, a single real fixed point. In particular, elliptic elements of G can only have traces 0 or ±1; the former are of order 2, the latter of order 3, and these are the only orders possible for elements of G. We shall consider that all groups of linear-fractional transformations act on H or on the real axis.
The points a, b, [member of] H are said to be G-equivalent (or simply equivalent, when G is understood) if Va = b for some element V [member of] G. By this equivalence relation H is partitioned into mutually disjoint equivalence classes or orbits
Gz = {Vz |V [member of] G}.
The concept of orbit leads to the fundamental region, on the one hand, and to the Riemann surface, on the other. These are both realizations of the orbit space H/G, defined as the set of distinct orbits of G.H/G is the space obtained by identifying points in H that are G-equivalent.
To realize the orbit space in H, select one point from each orbit and call the union of these points a fundamental set for G (relative to H). Since we wish to deal with nice topological sets, we modify this concept slightly and define a fundamental region RG to be an open subset of H which contains no distinct G-equivalent points and whose closure contains a point equivalent to every point of H. That fundamental regions exist for the groups of interest to us admits a simple proof (cf. Gunning, ch. I or Short Course, p. 57). Fundamental regions for Γ and Γ(2) are shown in the figures; notice that they are actually regions (i.e., connected), which is not required by the definition. There are, of course, many fundamental regions; in particular, V (R) is one if R is, where V [member of] G. The collection of regions {V (R)|V [member of] G} form a network of nonoverlapping regions which, with their boundary points, fill up H. Examples of these striking geometric configurations may be found in many books.
Let A, B be sets in which a multiplication of elements is defined. Write
C = AB
if the set of products {ab|a [member of] A, b [member of] B} is the set C. Write
C = A · B (4)
if for each c [member of] C we have uniquely c = ab, a [member of] A, b [member of] B. Thus, with C a group, A a subgroup, we have (4), where B is a system of right representatives
Theorem 1.Let G = H · A, where H is a subgroup of finite index in G. Then
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
is a fundamental region for H. ([bar.R] is the closure of R.)
The almost evident proof can be found in Short Course, Theorem 6D, p. 61f. This fundamental region may not be connected; however, connected fundamental regions do exist for all subgroups.
It is possible to select the fundamental region so that it has other desirable properties. The sides of such a fundamental region are arranged in conjugate pairs, the two sides of a pair being equivalent by a group element. These conjugating transformations generate the group.
For example, in R,Γ we regard the arc of the circle as consisting of two sides separated by the point i. Then the vertical sides are mapped into each other by S, the curved sides by T, where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
are standard notations. (Short Course, p. 37.) The vertices are defined to be points of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where two sides meet; they are arranged in cycles, each cycle being a complete G-equivalence class of points on the boundary of RG. Thus RΓ has the following cycles: {∞}, {i}, {e,]ITL2πi/3, eITL2πi/6}; while RΓ has the cycles: {∞}, {0}, {-1, 1} – see figure on p. 3. If one vertex of a cycle is fixed by a parabolic element P, the other vertices are also fixed points of parabolic elements, for V P V-1 fixes V α if P fixes α and V P V-1 is also parabolic. Then the cycle is called parabolic. Similarly for elliptic fixed points, and it is clear that every vertex of an elliptic cycle is fixed by an elliptic element of the same order, which is called the order of the cycle. It can happen that the vertices of a cycle are not fixed points at all; then we say the cycle is accidental(or unessential). Such a cycle can be avoided by a different choice of fundamental region, whereas this is not possible with fixed-point cycles. Obviously an elliptic cycle must lie entirely in H, whereas the vertices of a parabolic cycle are all real (including possibly ∞). The sum of the angles at the vertices of a cycle is 0, 2π, or 2π/k, according as the cycle is parabolic, accidental, or elliptic of order k > 1. In RΓ the cycle {∞} is parabolic, while the remaining cycles are elliptic of order 2 and 3, respectively. In RΓ the cycles are all parabolic. (Short Course, p. 39ff.)
If we identify points (necessarily on the boundary) of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] that are G-equivalent we obtain an orientable surface. Thus for RΓ we get a sphere with one point removed (one "puncture"); for RΓ we get a sphere with 3 punctures. The genus of the surface (i.e. the number of handles) is in both cases zero. It can be shown that this always happens: the RG of every subgroup G of finite index in Γ becomes, on identification of G-equivalent boundary points, a surface of genus g ≥ 0 with t ≥ 0 punctures (Short Course, ch. III, sec. 1). The integer g can be computed from Euler's formula applied to RG:
c - n + 1 = 2 - 2g (6)
where c is the number of cycles and 2n the number of sides. The number of punctures t is simply the number of parabolic cycles in RG.
Let us calculate the genus of Γ. Here n = 2 and there are 3 cycles; this gives g = 0. For Γ(2) we have n = 2 and 3 cycles; again g = 0.
These considerations tie in naturally with the Riemann surface of G. The orbit space R = H/G can be given a topological and, in fact, analytic structure that makes it a Riemann surface. There is a projection map
σ; τ -> Gτ
from H to R which identifies G-equivalent points:
σ0V = σ for V [member of] G, (7)
and which is a local homeomorphism except at the fixed points of G lying in H. Thus H may be regarded as a branched unlimited covering of R = H/G with the projection σ. Moreover, RG is a model of R and the genus of R is the same as the genus of RG as defined above (Short Course, ch. III, sec. 1). Finally σ is an analytic mapping from H to R except at the fixed points of G.
Poincaré introduced the area metric
dudv/v2, τ = u + iv (8)
which is invariant under all linear-fractional transformations with real coefficients and determinant +1. If we restrict ourselves to fundamental regions that are Lebesgue measurable, it is easy to show that they all have the same area, which is therefore a group invariant and is called the area of the group (Short Course, p. 50). The area under this metric of a circular arc triangle lying in the closure of H is known to be π minus the sum of its angles (Gauss-Bonnet formula). By triangulation RG we then find, in view of (6), that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
where |RG| is the area of RG and li is the order of the transformation fixing any point of the ith cycle (=1 if the cycle is accidental, ∞ if the cycle is parabolic). Thus |RG| is finite. C. L. Siegel has determined the minimum of the right member of (9):
|RG| ≥ π/21. (10)
This remarkable but completely elementary result was apparently unknown until the appearance of Siegel's paper in 1945 (Some remarks on discontinous groups, Ann. of Math.). For groups with parabolic, elements – all the groups G are in this class – we have a stronger result:
|RG| ≥ π/3. (11)
The minimum is attained by Γ. (Short Course, ch. I, sec. 5.)
From Theorem 1 we deduce that if [Γ : G] = μ, then
|RG| = μ|RΓ|, (12)
since |V (RΓ)|= |RΓ|.
2 Modular Forms
A modular form is an analytic function that has a certain type of invariant behavior on Γ or a subgroup of Γ. We recall that the symbol G is used to denote an arbitrary subgroup of finite index in Γ. The simplest behavior would be strict invariance:
f(Lτ) = f(τ) for all L [member of] G. (13)
However, this is too restrictive for our purposes. If we consider differentials rather than functions, we observe that a consequence of (13) is
f'(Lτ)d(Lτ) = f'(τ)dτ.
Differentials of higher "weight" would be obtained by taking powers; thus, with g = f'h we get
g(Lτ)(dLτ)h = g(τ)(dτ)h.
Now
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
if we remember that ad - bc = 1; the above equation is then
(cτ + d)-2h g(Lτ) = g(τ), L [member of] G, h [member of] Z. (14)
The well-known Weierstrassian invariants g,2g3 from the theory of elliptic functions satisfy (14) with 2h = 4 and 6, respectively.
To express (14) it is very convenient to introduce the stroke operator:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)
Here L has real coefficients and determinant ≠ 0; in our applications k = 2h is an even integer and is usually omitted from the notation. Equation (14) is then simply
g|L = g, L [member of] G. (16)
Observe that
g|L1L2 = (g|L1)|L2. (17)
We can also define, for constants α1, α2:
g|(α1L1 + α2L2) = α1g|L1 + α2g|L2. (18)
Moreover, we shall want g to satisfy some analytic requirements. It would be natural to insist that g be meromorphic, but in our applications only holomorphic functions arise. The second requirement concerns the behavior of g at real rational points (including ∞): we wish g to have a definite behavior as τ tends to such a point. We now make the following definition.
Let {G, -k} be the set of functions f(τ) such that (19)
(i) f is holomorphic in H.
(ii) for each matrix A [member of] GL+ (2, Z), i.e., with integral entries and positive determinant, f | -k A tends to a definite limit (finite or infinite) as τ = u + i v ->i∞ uniformly in every region Ea: |u| < 1α, v > α > 0; this limit is independent of u.
(iii) f | -k L = f for all L [mmeber of] G.
If r is a rational point, we define the value of f at r as follows: Let V be a matrix of GL+ (2, Z) such that V ∞ = r; define
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)
The existence of this limit follows from (19)(ii); that it is independent of V (provided V ∞= r) is easily shown. Thus imposition of (ii) guarantees that f has a value at all rational points. Note that
(f + g)r = (f)r + (g)r and (f|L)r = (f)Lr
The value (f)r may be infinite. Note that (f)r is not necessarily equal to f (r) = limτ ->r f (τ). For suppose f [member of] {Γ, -k} and suppose f (∞) = 1. Let r be finite and let V ∞ = r, V = (a b|cd). Then c ≠ 0. So f (V τ) = (cτ + d)k f (τ); letting τ -> ∞ we have f (r) = ∞. But (f)r = (f|V)∞ = (f)∞ = 1. However (f)∞ = f (∞), since we may take V = I.
We call f a modular form of degree -k (actually, a regular modular form). If k = 0, we speak of a modular function. Restrictions like (ii) are absolutely essential if the theory of modular forms is to have an algebraic character.
(Continues...)
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