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We recall Table 30.1 in Sect. 30.1 which displays numbers of eta products of weight 1 for some of the Fricke groups which will be inspected in the present section. [ABSTRACT FROM AUTHOR]

In the final two sections of this monograph we inspect eta products of weight 1 on Fricke groups of levels N=4pq for several pairs of distinct odd primes p and q. Table 30.1 displays the numbers of these eta products and of those for Γ0(4pq) for a few small values of p,q. [ABSTRACT FROM AUTHOR]

For primes p>q≥3 there are only two new holomorphic eta products of weight 1 and level p2q2, namely, η(q2z)η(p2z) and η(z)η(q2p2z). They belong to the Fricke group, and their orders at ∞ do not allow the construction of eigenforms. Thus our inspection of eta products in this section is confined to the Fricke groups of levels 4p2 for odd primes p. Table 29.1 displays the numbers of new holomorphic eta products also for the groups Γ0(4p2). For p≥7 the numbers for Γ∗ (4p2) are independent from p, and each of these nine eta products is a product of two simple theta series of weight ]> from Theorem 8.1. [ABSTRACT FROM AUTHOR]

For distinct odd primes p1,p2,p3, the only new holomorphic eta products of weight 1 and level N=p1p2p3 are ]> , ]> , ]> , ]> , and they belong to the Fricke group. We do not expect to find linear combinations of these eta products and some complementary functions which are eigenforms. [ABSTRACT FROM AUTHOR]

Here and in the following sections we will inspect eta products of some levels which have 8 or more positive divisors. A class of levels N with σ0(N)=8 is given by N=q3p where q and p are distinct primes. Table 27.1 shows the numbers of new holomorphic eta products of weight 1 for some values of q and p. Since some of these numbers are quite large, we restrict our diligence to eta products for the Fricke groups. [ABSTRACT FROM AUTHOR]

We recall that the non-cuspidal eta products of weight 1 for the Fricke group Γ∗ (12) were treated in Sect. 25.1. According Table 24.1 in Sect. 24.1 there are 158 more such eta products for Γ0(12). We start inspecting those with large denominators, working down to denominator 1. Here for the first time we meet non-cuspidal eta products with denominator 24. They form eight pairs of sign transforms, where one member in each pair has a non-zero value only in the orbit of the cusp 1, and the other member has a non-zero value only in the orbit of the cusp ]> . Four of these pairs combine to theta series on the field ]> , the other four combine to theta series on ]> , and all of these series are also Eisenstein series. We describe the results in the following two examples. [ABSTRACT FROM AUTHOR]

We devote two sections to the discussion of the large number of eta products of level 12 and weight 1. (See Table 24.1 at the beginning of Sect. 24.1.) In the first of these sections we deal with all the cuspidal eta products for Γ0(12) and, as an exception, also with the non-cuspidal ones for Γ∗ (12). Results for this Fricke group are contained in the table at the end of (Kahl and Köhler in J. Math. Anal. Appl. 232:312–331, 1999), where, however, no details on the characters are communicated. In this subsection we will provide more details, and we will add results for real quadratic fields. [ABSTRACT FROM AUTHOR]

From Sect. 21.1 we know that for primes p≥7 there are 9 new holomorphic eta products of weight 1 for Γ∗ (4p) and another 60 such eta products for Γ0(4p). Table 24.1 shows that the corresponding numbers of eta products are slightly larger for level N=4ω5=20 and considerably larger for level N=4ω3=12. In the present section we are going to discuss theta series identities for the eta products of weight 1 and level 20. [ABSTRACT FROM AUTHOR]

For primes p≥5 there are exactly four new holomorphic eta products of weight 1 for the Fricke group Γ∗ (2p), namely, ]> By Theorem 8.1, each of them is a product of two simple theta series. All of them have denominator 8 if pO1 mod 3, while for pO−1 mod 3 the denominators are 8 for the first and second, and 24 for the remaining two eta products. Some of the identities in this subsection are mentioned in (Kahl and Köhler in J. Math. Anal. Appl. 232:312–331, 1999). We begin with the discussion of the case p=5, where we will meet theta series on the fields with discriminants 40, −40 and −4. [ABSTRACT FROM AUTHOR]

In this and the following two sections we discuss eta products whose levels N=pq are products of two distinct primes p,q, whence the number of divisors of N is 4. In the present section we begin with the case of odd primes 3≤p∗ (pq). [ABSTRACT FROM AUTHOR]

Once more, we recall Table 21.1 in Sect. 21.1 with the numbers of eta products of weight 1 and levels N=4p. Now we discuss the primes p=11 and p=7. [ABSTRACT FROM AUTHOR]

There is exactly one cuspidal eta product of weight 1 for Γ∗ (68) with denominator 4. Its order at ∞ is ]> , and it is the sign transform of the function η(z)η(17z) which was treated in Example 12.10. Not surprisingly, now we get a similar result for theta series on ]> with characters which have twice the period of those in Example 12.10. Here again, one of the components is identified with a difference of two non-cuspidal eta products. [ABSTRACT FROM AUTHOR]

For each prime p≥7 there are exactly nine new holomorphic eta products of weight 1 for the Fricke group Γ∗ (4p), where five of them are cuspidal and 4 are non-cuspidal. In addition there are 60 new holomorphic eta products of weight 1 for Γ0(4p), 44 of them cuspidal and 16 non-cuspidal. [ABSTRACT FROM AUTHOR]

For primes p≥5 there are exactly 6 holomorphic eta products of weight 1 and level N=p2. The only new one among them is η(z)η(p2z). Since its order at ∞ is ]> , there is little chance to find complementary eta products for the construction of eigenforms which might be represented by Hecke theta series,–at least when we stick to level p2. The chances are improved when we consider η(z)η(p2z) as an old eta product of level 2p2, and indeed the function η(z)η(25z) will play its rôle in Sect. 20.3. [ABSTRACT FROM AUTHOR]

For primes p≥5 the only holomorphic eta product of weight 1 and level p is ηp(z)=η(z)η(pz). It belongs to the Fricke group. If the order at ∞ satisfies ]> then we can find complementary components such that a linear combination with ηp(z) becomes a Hecke theta series. For pϵ{5,7,11,23} the numerator of the eta product is one, ]> . Then ηp(z) itself is a Hecke theta series. These cases are known from (Dummit et al. in Finite Groups–Coming of Age. Contemp. Math. 45:89–98, 1985) and (Kahl and Köhler in J. Math. Anal. Appl. 232:312–331, 1999). The result for p=23 was discussed even earlier by van der Blij (Koninkl. Nederl. Akad. Wetensch. 55:498–503, 1952) and Schoeneberg (Koninkl. Nederl. Akad. Wetensch. 70:177–182, 1967). For p=5 and p=7 theta series identities involving real quadratic fields are known from (Kac and Peterson in Adv. Math. 53:125–264, 1984), (Hiramatsu in Investigations in Number Theory. Advanced Studies in Pure Math. 13:503–584, 1988). [ABSTRACT FROM AUTHOR]

In six sections we will discuss eta products of weight 1 with levels N=p2q where p,q are distinct primes. For these levels the number of positive divisors is σ0(N)=6. In the present section we treat the case that p is odd, and here most of the effort is needed for level N=18. Five more sections will be dedicated to the case p=2, that is, levels N=4q with odd primes q. We need so much space for the results in the case p=2 due to the fact known from Theorem 3.9, part (3), that the number of holomorphic eta products of a given weight increases when a prime in the factorization of the level is replaced by a smaller prime. [ABSTRACT FROM AUTHOR]

The number of divisors of an integer N is 6 if and only if N=p5 or N=p2q with distinct primes p,q, and it is equal to 7 if and only if N=p6 for some prime p. In this section we discuss eta products whose levels are p5 or p6. Other levels N with six positive divisors will be treated in the following sections. [ABSTRACT FROM AUTHOR]

Table 18.1 displays the numbers of new holomorphic eta products of level 6 and weights 1 and 2, specified according to their groups and denominators. The large numbers for Γ0(6), k=2 suggest that we confine our discussion to eta products of weights k≤2 for Γ∗ (6) and to those of weight 1 for Γ0(6). [ABSTRACT FROM AUTHOR]

So far we worked through the levels N with numbers of divisors τ(N)≤3. In Sects. 15, 16, 17 and 18 we will delve into the cases with τ(N)=4 or 5. These are the cubes and fourth powers of primes and the products of two distinct primes. [ABSTRACT FROM AUTHOR]

There are six new holomorphic eta products of weight 1 for the Fricke group Γ∗ (4). They are the sign transforms of η2(z) and of the five eta products for Γ0(2) listed at the beginning of Sect. 10.1. Therefore the representations by theta series are quite similar to those in Sect. 10.1. A minor difference is that we need larger periods for the characters. It is easy to verify the following result, which allows a comfortable construction of modular forms for the Fricke group Γ∗ (4). [ABSTRACT FROM AUTHOR]

For Γ0(3) and weight k=1 there are three holomorphic eta products, ]> The first one is cuspidal and belongs to the Fricke group Γ∗ (3), the others are non-cuspidal. Here we have an illustration for Theorem 3.9 (3): The lattice points on the boundary of the simplex S(2,1) do not belong to S(3,1), and two of the interior lattice points in S(2,1) are on the boundary of S(3,1). At this point it becomes clear that η(z)η(pz) is the only holomorphic eta product of level p and weight 1 for primes p≥5. The eta product η(z)η(3z) is identified with a Hecke theta series for ]> ; the result (11.2) is known from (Dummit et al. in Finite Groups–Coming of Age. Contemp. Math. 45, 89–98, 1985), (Köhler in Math. Z. 197, 69–96, 1988). [ABSTRACT FROM AUTHOR]

For a real number λ>0 the Hecke groupG(λ) is defined to be the subgroup of SL2(ℝ) which is generated by ]> and ]> . We mention the Hecke groups not because of Hecke΄s pioneering research (Hecke in Lectures on Dirichlet Series, Modular Functions and Quadratic Forms, Vandenhoeck & Ruprecht, Göttingen, 1983), but merely since three of them are conjugate to Fricke groups: Besides the modular group G(1)=Γ1 itself, we have ]> The Hecke group G(2) is also called the theta group since Jacobi΄s θ(z) is a modular form for G(2). Several of the results in Sects. 10, 11 and 13 are transcriptions of earlier research (Köhler in Abh. Math. Sem. Univ. Hamburg 55, 75–89, 1985), (Köhler in Math. Z. 197, 69–96, 1988), (Köhler in Abh. Math. Sem. Univ. Hamburg 58, 15–45, 1988) on theta series on these three Hecke groups. [ABSTRACT FROM AUTHOR]

Clearly, the only holomorphic eta product of weight k for the full modular group is η2k(z). Lacunary powers of the eta function have been studied by Serre [Glasgow Math. J. 27, 203–221, 1985] exhaustively. Here we will present theta series representations for some modular forms on the full modular group, including Serre΄s results on powers of η(z). [ABSTRACT FROM AUTHOR]

In this section we describe systems ]> of integral ideal numbers for all those imaginary quadratic fields ]> which will occur in the examples of the following sections. In the last subsection we will also describe ideal numbers for those few real quadratic fields that will be needed. [ABSTRACT FROM AUTHOR]

In Example 3.12 we learned that there are exactly six holomorphic eta products of weight ]> which are new for the levels 1, 2 or 4. In Sects. 1.1 and 1.2 we obtained series expansions for four of these functions. In a closing remark in Sect. 3.6 we explained that these expansions are simple theta series for the rational number field with Dirichlet characters. Now we derive similar expansions for the remaining two eta products ]> [ABSTRACT FROM AUTHOR]

For an explicit specification of a Hecke theta series for a quadratic number field K we need an explicit definition of characters on the groups ]> and ]> where ]> is a non-zero ideal in ]> , M is an ideal number for ]> , and ]> is a system of integral ideal numbers for K. Since ]> and ]> are finite abelian groups, they are isomorphic with direct products of cyclic subgroups. When we know generators of the direct factors then we can define a character by specifying its values on the generators. In almost all of the examples in Part II we will define characters in this way. For this purpose we need to know a decomposition of the groups into direct factors, and we need to know generators of the factors. [ABSTRACT FROM AUTHOR]

In 1920 Hecke (Math. Z. 6, 11–51, 1920, Math. Werke 14, 249–289) introduced a new kind of theta series. The corresponding Dirichlet series form a common generalization both of Dirichlet΄s L-series and of Dedekind΄s zeta functions. While Dirichlet΄s L-series are defined by characters on the rational integers, Hecke΄s L-functions involve characters on the integral ideals of algebraic number fields. The values of these characters at principal ideals depend on the values of the algebraic conjugates aν of a generating number a, and not just on the residue of a modulo a fixed period ideal. Therefore Hecke called his characters Grössencharaktere. We prefer to use the term Hecke character. We can find definitions and results on Hecke characters, Hecke theta series and Hecke L-functions in some textbooks; we mention (Miyake in Modular Forms, Springer, Berlin, 1989), pp. 90–95, 182–185, (Neukirch in Algebraische Zahlentheorie, Springer, Berlin, 1992. English Translation: Algebraic Number Theory, Springer, Berlin, 1999), pp. 491–514. Here we will reproduce relevant definitions and results, but we will not give proofs. [ABSTRACT FROM AUTHOR]

In Sect. 2.5 we obtained a bijection between the holomorphic eta products of level N and the lattice points in a closed simplicial cone ]> in ℝτ(N). Since η(mz)2k is a cuspidal eta product of level N and weight k for every m|N and every (integral or half-integral) k>0, the half lines from the origin through the standard unit vectors belong to the interior of ]> . Therefore, the first octant {X=(xm)mϵℝτ(N)ǀX≠0, xm≥0 for all m|N} belongs to the interior of ]> . [ABSTRACT FROM AUTHOR]

In Part II we will show plenty of examples of eta products or linear combinations of eta products which are Hecke eigenforms and which are represented by theta series with a Hecke character on some imaginary quadratic field. Our starting point for exhibiting these examples is a list of all holomorphic eta products of a given level N and weight k. The results in Sect. 3 say that we get this list when we list up all the lattice points in a certain compact simplex. Every single lattice point represents an interesting function, and we really need such a list. [ABSTRACT FROM AUTHOR]

By an eta product we understand any finite product of functions ]> where m runs through a finite set of positive integers and the exponents am may take any values from ℤ, positive or negative or 0. (Of course, an exponent 0 contributes a trivial factor 1 to the product, and therefore we may as well assume that am≠0 for all m.) Since the product is finite, the lowest common multiple N=lcm {m} exists, and every m divides N. [ABSTRACT FROM AUTHOR]

Throughout this monograph we use the notation ]> where z is a complex number. We define the Dedekind eta function by the infinite product 1.1 ]> The product converges normally for q in the unit disc or, equivalently, for z in the upper half plane ℍ={zϵℂǀIm(z)>0}. This means that the product of the absolute values |1−qn| converges uniformly for z in every compact subset of ℍ. The normal convergence of the product implies that η is a holomorphic function on ℍ and that η(z)≠0 for all zϵℍ. [ABSTRACT FROM AUTHOR]