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The construction of modular forms as products of transforms of the Dedekind eta function. (English) Zbl 0718.11017

For the Dedekind eta-function \(\eta\), a transform of level n in reduced form is any nonzero constant multiple of a function \(\eta ((a\tau +b)/d)\) where a,b,d are relatively prime integers satisfying \(ad=n\) and \(0\leq b<d\). An eta product is a finite product of transforms of arbitrary levels with arbitrary real exponents. Such eta products have appeared in many parts of Mathematics, for example, in the theory of Lie algebras and in connection with sporadic simple groups (“moonshine”). The author studies systematically those modular forms which are eta products. A modular form is only required to be meromorphic on the upper half-plane H and in the cusps, it may have any real weight and any multplier system, but it has to belong to some subgroup of finite index in the modular group \(SL_ 2({\mathbb{Z}}).\)
The main results are as follows. Theorem B: The reduced transforms of any fixed level n are multiplicatively independent. Theorem C: A modular form f is an eta product if and only if f has no zeros and no poles in H and if the orders of f in the cusps satisfy certain consistency conditions. Theorem D: The eta products which are mdoular forms on the principal congruence subgroup \(\Gamma\) (n) are precisely the products of reduced transforms of level n with arbitrary real exponents. Theorem E: The eta products which are modular forms on \(\Gamma_ 0(n)\) can be described explicitly. The description is complicated in general, but if n or n/4 is squarefree then any eta product on \(\Gamma_ 0(n)\) is a product of functions \(\eta (a\tau)^{r(a)}\) with \(a| n\) and uniquely determined real exponents r(a).

MSC:

11F20 Dedekind eta function, Dedekind sums
11F11 Holomorphic modular forms of integral weight
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