ABSTRACT

is the disjoint union of for all, where is the set of all roots of primitive second degree equations, with reduc^ed discriminant equal to k2m. We study the action of two Hecke groups—the full modular group and the group of linear-fractional transformations on. In particular, we investigate the action of on for finding different orbits.

Keywords

Quadratic Irrationals, Hecke Groups, Legendre Symbol, G-Set

1. Introduction

In 1936, Erich Hecke (see [1] ) introduced the groups generated by two linear-fractional transformations

and. Hecke showed that is discrete if and only if,

, or. Hecke group is isomorphic to the free product of two finite cyclic group of order 2 and q, and it has a presentation

The first few of these groups are, the full modular group having special interest for mathematicians in many fields of Mathematics, and.

A non-empty set with an action of the group G on it, is said to be a G-set. We say that is a transitive G-set if, for any in there exists a g in G such that. Let, where and m is a square free positive integer. Then

is the set of all roots of primitive second degree equations, with reduced discriminant equal to n and

is the disjoint union of for all k. If and its conjugate have opposite signs then is called an ambiguous number [2] . The actual number of ambiguous numbers in has been discussed in [3] as a function of n. The classification of the real quadratic irrational numbers of in the forms modulo n has been given in [4] [5] . It has been shown in [6] that the action of the modular group, where and, on the rational projective line is transitive. An action of, where and and its proper subgroups on has been discussed in [7] [8] .

invariant under the action of modular group G but is not invariant under the action of H. Thus it motivates us to establish a connection between the elements of the groups G and H and hence to deduce a common subgroup of both groups such that each of and is invariant under H^* and hence we find G-subsets of and H-subsets of or according as or and

for all non-square n. Also the partition of has been discussed depending upon classes modulo.

2. Preliminaries

We quote from [5] [6] and [8] the following results for later reference. Also we tabulate the actions on of and, the generators of G and H respectively in Table 1.

Theorem 2.1 (see [5] ) Let,. Then and are both G-subsets of.

Theorem 2.2 (see [5] ) Let. Then and are both G-subsets of.

Theorem 2.3 (see [6] ) If, then and are exactly two disjoint G-subsets of depending upon classes modulo 4.

Theorem 2.4 (see [6] ) If, then and

are both G-subsets of.

Lemma 2.5 (see [8] ) Let. Then:

1) If then if and only if.

2) if and only if.

Theorem 2.6 (see [8] ) The set, is invariant under the action of H.

Theorem 2.7 (see [8] ) For each non square positive integer, is an H-subset of.

3. Action of on

We start this section by defining a common subgroup of both groups and

, where, , and. For this, we need the following crucial results which show the relationships between the elements of G and H.

Lemma 3.1 Let and be the generators of G and H respectively defined above. Then we have:

1) and.

2) and.

3) and.

4) and.

5) and.

6) and. In particular and.

Following corollary is an immediate consequence of Lemma 3.1.

Corollary 3.2 1) By Lemma 3.1, since and so is a common subgroup of G and H where are the transformations defined by and.

2) As, , so is a proper subgroup of.

3) and.

Since for each integer n, either or for each odd prime p. So in the following lemmawe classify the elements of in terms of classes with 0 modulo p or qr, qnr nature of a, b and c modulo p.

Lemma 3.3 Let be prime and. Then consists of classes, , , , , , or.

Proof. Let be any class of. Then leads us to exactly three cases. If then exactly one of is and the other is qr or qnr of as otherwise and hence the class is one of the forms, , ,. If then and the class takes the form or. In third case if then so again. This yields the class in the forms or. Hence the result.

Lemma 3.4 Let and let be the class of of. Then:

1) If then has the forms, , , , , , , , , , or only.

2) If then has the forms, , , , , , , , , , or only.

Proof. Let be the class of with. As so if

then according as or. Thus we have, if and, if. If then, so we get, , , , , , , , or only. This proof is now complete.

Lemma 3.5 Let and let be the class of of. Then:

1) If then has the forms, , , , or only.

2) If then has the forms, , , , or only.

Proof. The proof is analogous to the proof of Lemma 3.4.

Note: If then, and are three classes of in modulo 2. If n is an odd then three classes of are, and modulo 2. These are the only classes of if. But if then is also a class of and there are no further classes. These classes in modulo 2 of do not give any useful information during the study of action of on except that if then the set consisting of all elements of of the form is invariant under the action of the group G. Whereas the study of action of H^* on gives some useful information about these classes. The following crucial result determines the H^*- subsets of depending upon classes modulo 2. It is interesting to observe that

splits into and in modulo 2. Each of these two H^*-subsets further splits into proper H^*-subsets in modulo 4.

Lemma 3.6 and are two distinct H^*-subsets of depending upon classes modulo 2.

Theorem 3.7 and Remarks 3.8 are extension of Lemma 3.6 and discuss the action of H^* on depending upon classes modulo 4. Proofs of these follow directly by the equations

, and classes modulo 4 given in [6] .

Theorem 3.7 Let n be any non-square positive integer. Then splits into two proper H^*- subsets,. Similarly splits into two proper H^*-subsets and .

Remark 3.8 1) Let. Then and are H^*-subsets of. In particular if, then and are H^*-subsets of. Whereas if, then, , and are H^*-subsets of. Specifically, ,.

2) As we know that if and are even, then must be even as. If, then and.

3) If, then or is empty according as or. As we know that if n and c are even, then a must be even as. However, are proper H^*-subsets of depending upon classes modulo 4.

Lemma 3.9 Let n be any non-square positive integer. Then and are distinct H^*-subsets of an H-set.

Proof. Follows by the equations and vice versa. Hence is equivalent to.

Clearly where denotes the set of all ambigious numbers in

(see [8] ).

Remark 3.10 1) Each G-subset X of splits into two H^*-subsets and and.

2) Each H-subset Y of splits into two H^*-subsets and.

3) Each H-subset Y of, splits into two H^*-subsets and.

4) Each H-subset Y of, splits into two H^*-subsets and.

Theorem 3.11 a) If A is an H^*-subset of or, then is a G-subset of.

b) If A is an H^*-subset of, then is an H-subset of or according as or.

c) If A is an H^*-subset of, then is an H-subset of for all nonsquare n.

Proof. Proof of a) follows by the equation.

Proof of b) follows by the equations or according as or.

Proof of c) follows by the equation.

Following examples illustrate the above results.

Example 3.12 1) Let. Then but. Also but. Similarly whereas. Also,. So has exactly 4 orbits under the action of H whereas splits into two G-orbits namely,.

2) splits into nine H-orbits. Also and . Whereas splits into four -orbits namely, and. (see Figure 1)

Theorem 3.13 Let p be an odd prime factor of n. Then and are two H^*-subsets of. In particular, these are the only H^*- subsets of depending upon classes modulo p.

Proof. Let be the class of. In view of Lemma 3.3, either both of

are qrs or qnrs and the two equations,

fix b, c modulo p. If then

or according as or. similarly for. This shows that the sets and are H^*-subsets of depending upon classes modulo p.

The following corollary is an immediate consequence of Lemma 3.6 and Theorem 3.13.

Corollary 3.14 Let p be an odd prime and. Then splits into four proper H^*- subsets depending upon classes modulo 2p.

Proof. Since implies that. This is equivalent to congruences and. By Theorem 3.13, are H^*-subsets and then, by Lemma 3.6, each of and

further splits into two H^*-subsets, , and.

The next theorem is more interesting in a sense that whenever, is itself an H^*-set depending upon classes modulo p.

Theorem 3.15 Let p be an odd prime and. Then is itself an H^*-set depending upon classes modulo p.

Proof. follows from Lemmas 3.4, 3.5 and the equations and given in Table 1.

Let us illustrate the above theorem in view of Theorem 3.4. If, then the set

is an H^*-set.

That is, is itself an H^*-set depending upon classes modulo 3. Similarly for.

Theorem 3.16 Let p be an odd prime and n is a quadratic residue (quadratic non-residue) of 2p. Then is the disjoint union of three H^*-subsets, and depending upon classes modulo 2p.

Proof. Follows by Theorems 2.6, 2.7 and 3.15.

The following example justifies the above result.

Example 3.17 Since, then splits into these three H^*-subsets, ,.

The next theorem is a generalization of Theorem 3.13 to the case when n involves two distinct prime factors.

Theorem 3.20 Let and be distinct odd primes factors of n. Then, , and are four H^*-subsets of. More precisely these are the only H^*subsets of depending upon classes modulo.

Proof. Let be any class of with. Then implies that

(1)

This is equivalent to congruences and. By Theorem 3.14, the congruence gives two H^*-subsets and

of. As, again applying Theorem 3.13 on each of and we have four H^*-subsets, , and of.

References