INTRODUCTORY REMARKS ON COMPLEX MULTIPLICATION

Complex multiplication in its simplest form is a geometric tiling property. In its advanced form it is a unifying motivation of classical mathematics from elliptic integrals to number theory; and it is still of active interest. This interrelation is explored in an introductory expository fashion with emphasis on a central historical problem, the modular equation between j(z) and j(2z).

We begin with an observation scarcely requiring ollege level mathematics: "If a room has a rectangular floor plan in which the ratio of its sides r is 2, then it may be partitioned into two rectangular rooms by a wall down the middle of the longer side (see Fig. la) in such a fashion that each half has the same shape as the original room (seen by using a 90 rotation)." Clearly this is a unique phenomenon as stated.Only that ratio r 2 will still be present after division into two halves (2/r r), and a (nonrectangular)   parallelogram-shaped room could not permit this kind of a subdivision (since its angles are not all equal).
Yet we can construct some kind of nontrivial parallelogram generalization, ann once we do this there is no place to stop short of describing a major part of pure mathematics'.What is involved is a synthesis of number theory, algebra, analysis, ana topology which now all comes unaer the heaaing of complex multiplication.Our purpose is to offer a minimal escription of this phenomenon using only "generally known" results.
We replace the rectangle by a parallelogram with a perloalc structure (torus).
This means that the parallelogram may now be subaiviae into pieces which may be move an reassemble accoralng to the group of vectors generate by the sies of the parallelogram.For instance, in Fig. Ib, the square A + B + C + D (on the right) can be re- assembled as the rectangle A* + B* + C + D, where A* (or B*) is the right-hand (or downward) translate of A (or B).This new rectangle A *-+ C + D + B* can be cut in half to produce D + B* and A* + C, each of which has half the size but the same shape as the original (as seen by a 45 rotation).
If this looks too obvious, consider the parallelogram on the right in Fig. ic.It is made up of isoscles triangles A and D of sides I, 2, 2 and scalene triangles B + C and E + F of sides I, 4r2, 2. Thus the parallelogram A + B + C + D + E + F is equivalent to A* + B* + C* + D + E + F and each half (namely A* + B* + F and C* + E + D) is similar to the original (as seen by a rotation of arccos i/8 69 ...).
When we introduce complex numbers we see that these are essentially the only shapes for division of a parallelogram "modulo translations" into two equal similar parts, and we also see why the configurations are described as "complex multlplica- tions of norm two".
We consider two elements of K* (nonzero complex numbers) which are independent over IR, namely Wl and w2 (with a nonreal ratio).We define this ratio as T 2/i Im > 0 (T 6 H+). (3.1) Thus the ordering of Wl and 2 is chosen to make T lie in the upper-half plane H+ Using Wl and 2 we generate the lattice L, which is thought of either as a set of points or as an abelian group of translations.We write e [i'2 {nlwl + n22 n l,n 2 6 7.] (3.2) (or the X-module of rank two generated by Wl and 2 in ).
The lattice L has many equivalent bases, i.e., it may be written as e [I,2] (3.3a) where, symbolically, k A for a matrix A in PSL2(.) the modular roup,(the matrices A and -A are identified).In detail, kl alll + a122 aij 6 Z, A (aij) k2 a21I + a22w2 alla22 a12a21 1 (3.3b)The determinant of A is +i (not +i) to preserve ordering, i.e., so that 2/I E H+ also The changes of bases are generated by the operations indicated as follows: (3.4) The group PSL2(.) is more complicated than L, and nonabelian. (For further tails see Jacobson [12] or Gunning [9]).
The advantage of complex numbers comes with the concept of "shape" (or euclidean similarity).It can be defined as the equivalence class of lattices [L} {L I] related by L--LI, E *, (3.5a) which denotes the iattice attained by multiplying each element of L by Thus, in the symbols of (3.2) and (3.3a) Thus each L is equivalent (using i/wl to L I [1,7], Im 7 > 0, (3.6a)where many different T (e.g., , 0 may be used, all related by PSL2(T.)(see(3.3)), [1,7] [i,0] 0 (all + a12)/(a217 + a22) (3.6b)We finally note the inclusion symbol L c L (3.7a) 0 i for L 0 a sublattice of L I Actually, when L 0 is viewed as an additive subgroup LI, the index ILl/L01 is easily interpreted as the (finite) ratio of areas of t, parallelograms formed by the basis vectors of L I and L0o Thus when similarity occurs, we easily recognize Ie/ El II 2 (3.7b) lhus we now verbalize Fig. 1 as the display of lattices L and complex multi- with norm given by IL/ L! {I 2 2   (3.8)We seem to be doing even better; we can arrange to have For example, in Figs.labc (respectively we can recognize thi by using (3.4) as fol- lows: (In each case, the analogues of (3.9) are innicate by a*).
We now consider R[L}, the set of all multipliers of L, i.e., R{L} {p 6 IP L = L} We include p 0 in the sense that OL is trivially in L. [el, (e ann e, above, have the same multipliers).Thus R[e} is callen the multi- plier rin of each e or the class [e}.If R[e} Z, it is seen to contain a complex element, which performs complex multiplication. (Real multipliers must be integers by innepennence of the basis vectors of L).
THEOREM 4.3.Complex multiplication occurs in a lattice L [ml,2] if and only if the ratio 2/i (= m) is a quanratic sur satisfying an equation over  (4.4) normalizen so that a > 0, gcn (a,b,c)   i.Then the multiplier ring is generated by a quadratic integer , i.e.,

R{e} z[] (4.5)
Here satisfien a monic equation of integral trace t and norm n, and, more important, with the same niscriminant n, 2 t + n 0, n t 2 4n (< 0).T b21 + b22T where bij 6 Z, (but bllb22 b12b21 I 12).We easily recognize as an "eigen- value" and [I,T] as an "eigenvector", leading to equations of type (4.4) for T, and monic for .But here is not necessarily a generator of the ring RILl, (it may be 2, etc.).The main difficulty is showing that RILl has a generator which necessarily leads to the s.ame discriminant d. (In practice this is done by quadratic forms see the next section).Note the dependence (4.9) We can show (4.9) by an explicit choice of generator i + m, (m Z) defined by the cases If complex multiplication of L occurs by (6 R[L] with norm I 12 n, then a basis of L can be chosen so that where m is the maximum integer of for which /m 6 R [L]   For proof, consider the elementary divisor form of the mapping L -L.Note the factor m by itself is a real multiplication, and m 1 for the interesting cases.
We can now proceed to enumerate all complex multiplications.The first few are easy because they take place in lattices [i,] whose ring is just Z[].For norm we have units for d -4 and -3, namely i, The complex multiplications of norm 2 are in (3.10abc), and have discriminants d -8, -4, -7 respectively.It can be verified that for a given norm n, only a limited number of occur in (say) (4.6), (since t 2 4n + d < 4n).

CLAS S NUMBER.
We define h h(d), the class number of d as the number of inequlvalent lat- tices LI,...,L h with complex multiplication and the common dlscriminant d d{Li].
Each R{Li] is the same -[i from (4.10ab), so we can always write L I [i,i] the so-called principal lattice, and we call [L I] the principal lattice class.
Historically lattices were treated by Gauss (1800) by a correspondance with hi- nary positive quadratic forms (n l,n2) This is done by writing L [w l,w2 with e (n l,n2) tlnl01 + n2212 (5.2a) where t E I is chosen so that takes the form, in integers of ., (nl,n2) an + bnln2 + cn a > O, gcd(a,b,c) I, Borel etc. [2] and deSeguier [15]).
We bypass a wealth of technique to concentrate on the case (5.1ab) for d -20.
For instance, I represents i (and is indeed called prin_cipal accordingly, while + n2)2 + 5n 2 for n I and n 2 in .
2 does not i.e. 22(ni n2) (2nl (The term principal ideal arose eviously in this context; L I is principal in . -5] while L 2 is not).
This result is the "Gauss-Heegner-Stark Theorem", (see Stark [18]).It is a major illustration of the use of transcendental functions for algebraic purposes.
We shall merely define its man analytic tool, the modular invariant j[L} with emphasis on the more manipulatively available aspects, rather than venturing into topology and becoming absorbed in a different world, (e.g., of forms an Riemann Surfaces, see Gunning [9]).
By definition, we are demanding that the modular invariant j[L] be analytic in T (E H+), if [I,T] E [e}.Thus we ask for invariance under PSL 2 (-), see (3.6b), a i T + a I j[L] j(T) j a I'' 2 a ., (6.1) We know from the generator properties in (3.4), that it suffices to show j(-1/T) To form this invariant, we start for symmetry's sake with L [l,m2 where the * denotes the omission from the sum of (nl,n2) (0,0). (The strange constants 60 and 140 are explained by (8.3) below).Now g2 and g3 depend on 3 2 L but not on {e]; nevertheless, g2/g3 depend only on [L] (or on ml/m2 With additional strange constants introduced we write j j{e] j(T) 1728g /(g23 27g 3 (6.5)THEOREM 6.6.The modular invariant j(T) satisfies (6.1) and has the further properties that (a) j(T) is analytic (free of poles) when Im T > 0, (in particular 3 2 g2 27g3 0), and j{L} is biunique onto (b) by the invariance under T --T + i, J(T) is a Laurent series in q exp 2 iT convergent for Im T > 0, and with integral coefficients starting with j(T) I/q + 744 + 196884q + 21493760q 2 + 864299970q 3 + (6.7) (c) any other analytic function of T invariant under PSL 2 (.) and with polar bounds at (< constlql "m, m 6 ), is a polynomial in We do not attempt the proof here, although part (c) is particularly the key to the connection with topology and Riemann Surfaces.We note the uncanny accuracy of (6.7) in computing j for the multipliers of norm 2, namely, jr-2) 8000 20 3, j(i)= 1728 12 3, j =-3375 =-153. (6.8) In addition in reference to (5.1ab), for d =-20, we compute jr-5)= (50 + 26r5) 3 jl + /-5 (50-26r5) 3 2 (6.9) which should further reassure us that ILl] [e 2].It may seem precarious, in (6.9), to guess irrationals from decimals, but the sum and product of the entries in (6.9) are integers.
THEOREM 6.10.For every discriminant d(< 0), let I denote the generator in (4.10ab).Then the value of j(l is an algebraic integer of degree h over the field k 0 0rd) (i ). (6.10a) It generates a field over (of degree 2h), namely, K k0(J(l)) (6.10b) which is abel ian over k 0. The h conjugates of j(l over k 0 are J[Li} (i l,...,h), the various classes of discriminant d.Thus from the fact that j(l is real, it is a rational integer precisely when h i, (see (5.4)).
This is part of a startling result in number theory of Weber (1890) (actually, "ring class field theory").We need the further definition that a prime p in K if for very element y of K the defining equation (over must, on reduction modulo p, contain only linear factors, possibly repeated.(One linear factor implies all factors of the defining equation are linear since K/ is normal.We could even assume is restricted to integers whose defining equation has no multiple factors).In the most familiar case (see Cohn [3]) we say that p splits in k 0 rd) exactly when d is a quadratic residue of p,(d/p)= I, (excluding those p which ivide d).The result is as follows: THEOREM 6.11.If the prime p does not divide d, then the condition for p to split in K, see (6.10b), is that we can represent p by the principal quadratl.Thus for h(d I, (6.12ab) follows exactly when p satisfies (d/p) I.
(6.12a) (6.12b) Weber [19] "almost" constructed all fields relatively abelian over the base field k 0 by using j(), but a later modification of Fueter [6] and Hasse [I0] was required to do it.(Kronecker had called for such a construction as his "Jugendtraum").Hilbert saw the phenomenon of a transcendental generator of algebraic fields and formulated his famous "twelfth problem" (see Langlands [14]) to investi- gate it further.Instead of describing the enormous complications to which this has lead, we retreat to the complex multiplications of norm 2 to show how they affect the j-function.
Before evaluating F2(X,Y) we note the symmetry in X and Y, because of the reciprocal ole of Jl j(2v) and J2 j(v/2).The symmetry might worry us because F2(J(2v) j(v)) 0 and F2(J(2v) j(4v)) 0, leaving the impression that J(4v) is a conjugate of J(v).To explain this "paradox" requires a careful accoun- ting of the Galois group permutations of the Ji (seen in (7.2abc) to be 3-dihedral).

ELL IPT IC INTEGRALS.
The classic way to forge a relation between j() and j( 2) is to find two elliptic integrals, one of which has periods in a lattice belonging to (e.g., L [l,W2 ]) and the other of which has periods in a lattice belonging to 2, (i.e., L' [wi,22]).Incredibly, the relations between such integrals are manag- eable by "undergraduate calculus".We use the Weierstrass method of starting with the lattice L [I,2 of periods and constructing elliptic functions with these periods.We start with (6.Then all elliptic functions with period lattice L are the field (z,w), where, (disdaining to discuss convergence), we set w p'(u) -2/u 3 -2 * I/(u )3.
Next, using the special coefficients in (6.4ab), we find Thus, for any constant ( 0), du is an elliptic integral of the first kind (always finite) and its periods form the lattices in the class [L}, as varies.
3 2 The polynomial P(z) 4z 3 g2 z g3 has three distinct roots (since g2 2/g3 0 from Theorem 6.6a).Hence a typical period of fdu itself is e e 2 2 2 u 2 =/e() <8.5) z=e I e I where, P(z) is now a cubic (and later on a blquadratic) with e I and e 2 as roots, (for a cubic, also behaves like a root ).The factor 2 in (8.5) comes from the fact that a period path does not really join two roots, but rather it encircles them so that the value of /P(z) can return to its original sign.Then the period path contracts to a doubled value.
To manipulate du it is necessary to change it to the Legendre form so du const dr, where dv dz/(l Ze)(l 'k 2Ze), k + I. (8.6a) We can do this by a linear transformation z (z + 8)/(vz + 6), ,8,,6 6 , 6 8 o. (8.6b) Fortunately, this transformation need not be specified All we need note is that we are transforming the roots el, e2, of 4z3 g2 z g3 into those of (I Z2)(I k2Z2), namely, i, -i, l/k, -i/k (8.b)We choose the right value of k for this by the cross-ratio.The cross-ratio of rl, r2, r3, r 4 is defined as k (r I r 2)(r 3 r 4)/(r I r 4)(r 3 r 2).
(8.16a) (8.16b) The modular equation (7.5) (now in X + Y and XY) emerges from a hand-calculation as the condition for a quadratic and a cubic polynomial (in t) to have a common root.
9. SOME FOLKLORE AND HISTORY.
The discovery of complex multiplication is connected with a legend which typically portrays C. F. Gauss as always the Ubermensch and never the Munchhausen in his claims to prescience and omniscience.At the age of 14, in 1791, Gauss in-  vented the arithmetic-eometric mean (agm), as a mere curiousity.Here one starts (9.2) Now, eight years later, in 1799, Gauss [8] calculated dx/vrl-x and naturally 0 took the ratio to its cognate dx TT/2 only to recall that this ratio 0 agreed to ii decimal places with the reciprocal of agm(l,2).He justifie this by proving that (under (9.1)), /2 de/a'-2 cos2e+b,2 sinZe as it would follow by iterating Tn(a,b), while a and b approached a common limit.
In terms of the integrals of type ($.6a), we can set sin lemniscate, i + r 2cos in polar coordinates.Gauss prove (9.4), essentially, by using the undoubled period in (.10ab).The doubled period occurred even more strongly in the form of theta-function identities (akin to the modular equation).
Remarkably, Gauss [7] did not publish this work until 1818, when he related it to the problem in astronomy of approximating the perturbation of a slow planet oy a fast planet, by replacing the fast planet's orbit by an elliptical ring.Meanwhile, in 1795, J. Landen in Cambridge had essentially produced the integral transforma- tion of (8.10ab), but presumably not with the modular interpretation.
There is a conflicting view that the credit for complex multiplication must be reserved to N. H. Abel [i] who first used period lattices and complex multipliers, but only in 1828 (see Cohn [4]).Thus it would be within Abel's scope to show complex multiplication by (i + i) by presenting the integral transformation of Gauss and Landen in the form of the statement that the differential equation ,2) z -2i/(z' I/z (9.6b) (Some minor changes of variable are required).Euler had previously shown in I51 how to produce a (real) factor of 2 (rather than i + i), but C. L. Siegel [16]   credits.Fagnano who found such a formula as early as 1718: In this century, Hecke [II] attempted to extend Weber's Theorem (6.11) to other fields, but this led mostly to a theory of modular functions of several variables.
Weber's original program is still being pursued on a much more abstract level by G. Shimura [17], R. Langlands [14], and others, (but it can scarcely be given a status report here).

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The details are omitted).The forms lle in h equivalence classes under PSL2(Z on n I and n2, which correspond uniquely to the classes [L i].(For etails, see nln2 (d I)n22/4, d -= i (mod 4).
the order (a-b)2 so the iteration of T easily converges as n--to lira Tn(a,b) (M,M), M agm(a,b).
ACKN(YLEDGMENTS.Based on research supported by NSF Grant MCS 7903060.Earlier versions of this worl.were presented at the University of Copenhagen in January 1977 under a visit sponsored by the Danish National Research Council.BIBLIOGRAPHICAL NOTEAn excellent exposition from an older vantage point was given by G. N. Watson in the Mathematical Gazette I(1933)5-I under the title "The Marquis and the Land- agent, A Tale of the Eighteenth Century".(He refers in the title to Fagnano and Landen, respectively).
L is closen unner annition ann subtraction.From this, R[L} Z. Further- more, as the notation innicates, R[L] really epenns on the equivalence class of since ].We