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Explicit Construction of the Hilbert Class Fields of Imaginary Quadratic Fields with Class Numbers 7 and 11

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Explicit Construction of the Hilbert Class Fields of Imaginary Quadratic Fields with Class Numbers 7 and 11 Explicit Construction of the Hilbert Class Fields of Imaginary Quadratic Fields with Class Numbers 7 and 11 Erich Kalto fen* University of Toronto Department of Computer Science Toronto, Ontario MSS1A4, Canada and Noriko Yui* University of Toronto Department of Mathematics Torontol Ontario MSS1A1, Canada Extended Abstract In this note we summarize the progress made so far on using the Com- puter Algebra System MACSYMA [10] to explicitly calculate the defining equa- tions of the Hilbert class fields of imaginary quadratic fields with prime class number. Our motivation for undertaking this investigation is to construct rational polynomials with a given finite Galois group. The groups we try to realize here are the dihedral groups Dp for primes p. These groups are non- abe]Jan groups of order 2p and are generated by two elements ~= (1 23..-p) andT= (1)(2 p)(3 p-l)-.-( p+l~- P+32 " with the relation ~0~r = e -1, as subgroups of the permutation groups of degree p. These groups are solvable and thus can be realized as Galois groups. The problem is to construct, for a given prime p, an integer polynomial with Galois group Dr . 1. C. U. Jensen and N. Yui have found the following effective characteriza- tion for polynomials to have Galois group Dr . Theorem (cf.. Jensen and Yui [7, Theorem II.l.2]): Let f (x) be a monic integral polynomial of degree p, where p is an odd prime. Assume that p =- 1 modulo 4 and that the Galois group of f is not the cyclic group of order p (resp. assume that p =- 3 modulo 4). Then necessary and sufficient conditions that the Galois group of f is Dr are: * This research was partially supported by the National Science and Engineering Research Council of Canada under grant 3-643-126-90 (the first author) and under grant 3-661-i14-30 (the second author). First author's current address: Rensselaer Polytechnic Institute, Department Mathematical Sciences, Troy, NewYork~ 12181. 311 (1) f is irreducible over the ring Z of integers. (2) The discriminant of f is a perfect square (resp./s not aperfect square). (3) The polynomial g(x) = 1-~l~i<j~p(x-a~-aj), a~ being the roots of f, which is of degree p(p-1)/2 and has all integral coefficients, decomposes into a product of [p-1)/a distinct irreducible polynomials of degree p over Z. [] Given an integral polynomial of degree p, it is quite easy to test whether conditions (1) - (3) are satisfied. Both the computation of the discriminant of f and that of the polynomial g ,can be accomplished by resultant calculations. The exclusion of the cyclic group of order p in the case thatp =- 1 modulo 4 may be more involved but it is, for example, sufficient to establish that f does not have p real roots. For p = 3, 5, and 7 polynomials with Galois group Dp are known for at least a century (cf. Weber [12, Sec. 131]). Unfortunately, extensive search for polynomials of degree 11 satisfying conditions (1) - (8) has not yet produced even one such polynomial. This is, to some extent, not surprising since the polynomial g will, for randomly chosen coefficients, almost always be irreducible due to ghe Hilbert irreducibility theorem. In order to construct such polynomials we therefore, at the moment, have to rely on tile Hilbert class field theory. We shall briefly sum- marize the theoretic background of our computations. 2. We consider an imaginary quadratic number field Q(~/m) with discrim- inant d over the fieldQof the rational numbers. Let ax 2 + bxy + cy 2, a > 0, GCD(a, b, c ) = 1, be a positive definite primitive quadratic form with discrim- inant d = b 2 - 4ac The integral matrix I~' with determinant a~ - 7fl = 1 transforms the quadratic form by replacing x by ax + fly and y by 7x + ~y into an equivalent one of the same discriminant g. The class number h(d) of Q(~/m-) is equal to the the number of such defined equivalence elasses of posi- tive definite primitive quadratic forms of diseriminant d. A unique reduced form for each equivalence class can be selected with -a <b <--a <c or O~b <_a =c. These conditions imply that Ib I -- x/Id I/3 and hence h(d) is finite. Now let SL2(Z ) be the modular{[o0] group: } SL2(Z) = c d t a, b, e, d EZ, ad-be = l , and let H denote the upper half complex plane: H= ~z =z +iy cCly>0l, where C is the field of complex numbers. SL2(Z ) acts on H by f~ b 1](z)- az +b [c d cz +d " 312 A fundamental domain F for SL2(Z ) in H is defined to be a subset of H such that every orbit of SL2(Z ) has one element in F, and two elements of F are in the same orbit if and only if they lie on the boundary of F. Then F is given by the set F= [z=x+iy ccl Izl->l, Ixl--< I- We now introduce the elliptic modular j-invariant. For each complex number z with non-negative imaginary part, let q = e 2n~z and let E4(z) = 1 + ~40 ~ aa(n)q n, o3(~) = ?, ~3 W,=I ~>0 Furthermore, let v(z)=q24 f[(i_q~)=q24 i+ (-iF(q 2 +q ~ ). The j-invariant j (z) is defined as j(z) : I n(~) 8 J " It is well-known that j (z) satisfies the following properties: (i) i(i) = 1728, /((it + g~/8)/2) = O, (ii) j(x + iy) and j(-x + iy) are complex conjugates for any ±x + iye F, and (iii) j(q) = 1 + 744 + 186884q + 21493760q e + 864289970q 3 + • • " . q 3. The following theorem now shows how to construct an integral polyno- mial with dihedral Galois group of prime degree. Theorem (cf. Deuring [2]): Let Q(~/m-) be an imaginary quadratic field with discriminat d, and with class number h(d) = p, p an odd prime. For each reduced positive definite primitive quadratic form a~x 2 + b~xy + c~y 2 of discriminant d, 1 -< /c ~p, let 9 k = (-b~ + "J-d)/(2ak) be the root of ak92 + b~ + c~ = 0 belonging to F. Furthermore, let the class equation /=Id be defined as HAx) = II (~ - i(~)) ~=1 Then H d(x) is an irreducible integral polynomial whose Gatois group over Q is the dihedral group D;. [] 4. We constructed H~(x) for selected imaginary quadratic fields Q(~/m) with -m a prime and h(d) = 7 or 11. First we wish to make some comments encountered during our calculations. In all cases we knew the class number in advance. Therefore it was quite easy to calculate the ~m, 1 ~ ]c _ p. Indeed, 313 for Ibt*l - a/, < c/~, we get two roots "Ot, = (=f:b~ + "4-d)/(2a~.), and for 0 < b k -< at* = c~, one root xgt* = (-be, + -,/-d)/(Eat*), belonging to F. Using the above mentioned properties of j we only had to evaluate j (gt*) for (p + 1)/2 different values of z~/~. The evaluation of each j(gt*) was done to high floating point pre- cision. We experienced that the Taylor series of 3" evaluated at q converged extremely slowly. Therefore we evaluated the Taylor series of E 4 and 77 separately at q, then raised the value zT(q) to the eighth power, divided E4(q) by this result, and finally raised the quotient to the third power. This process yields j (q) to high precision fairly quickly. In each case there were two parameters to choose: The floating point preci- sion and the order of the Taylor expansions. We decided to choose the same order for both E 4 and 7/. The constant coefficient of each polynomial turned out to be the one of largest size. Therefore we chose the floating point preci- sion typically 20 digits more than the number of digits in that coefficient. In all cases we then could read off the correct corresponding integer from its approximation. It turns out that the constant coefficient H a(O) must be a per- fect cube. Verifying this condition proved to be a valuable test to see whether the order of the Taylor approximation was high enough. If not, we incre- mented the order by 5 and tried again. A further confirmation for the correct- ness of all coefficients is to factor both Ha(O ) and the discriminant A(Hd) of H a both of which surprisingly have only small prime factors. A full explanation for this phenomenon has been found only very recently by ]~. Gross and D. Zagier. With their permission, we state a version of their theorem best suited for our discussion. Theorem (Gross and Zagier [3]): Let q be a prime. For a positive integer n e N such that (n) # + 1, define the function Fq (n) by with k, r i > 1 and n i _~ 0, G(~) = ifr~ =L• l2,,-1 ~,~-~ • • l~ *'-1 t where (~) = --1 with k~ - 1, s ~ 3 andt EN. (a) Let Q(a/m-), rn < 0 and -m a prime -= 3 (rood 4), be an imaginary qua- dratic field of discriminant d and of class number h(d) = h, an odd prime. Let Ok(x,y) = al, x 2 + bt*xy + ct*y 2, at, > 1, bt* > 0, /c = 1, 2 ..... (h-1)/2be the reduced positive definite primitive quadratic forms of discriminant d associ- ated with Q(n/m-). Let Ha(z ) be the class equation of Q(~lm-).
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