Infinite Products of Automorphic Forms

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The research project has been pursued mainly on number theory of automorphic forms, especially on the infinite products of modular functions. Details will appear elsewhere.1.Concerning the Fourier expansion of the modular invariant j(z), there is so-called Rademacher-Petersson's formula which gives the explicit values of the coefficients by some infinite series containing Kloosterman sums and Bessel functions. In some cases of higher level Γ_O(l) for suitable l's, it can be shown that a very similar formula holds on each Hauptmodul F_l(z) which has its own typical infinite product expansion. Their coefficients are given by quite distinctive subseries of one of j(z). We tried to know an intrinsic meaning of this phenomenon and reached another fact, which is expressed rather heuristically as followsj(z)〜Σ__<σ∈Γ_∞Γ(1)>q_σ^<-1>, F_l(z)〜Σ__<σ∈Γ_∞Γ_0(l)>q_σ^<-1>, q_σ=exp(2πiσ(z))However these coefficients given by such subseries are integral only when l=2,3,5,7,13, and its reason or the mechanism still remains to be unveiled in some days.2.We also worked on the problem of modular transformation formulas of the logarithm of Dedekind's eta function, especially its additive constant. This is a very classical and essential problem but is not completely clarified yet. Its investigation has still great significance, because not only the eta function is the most origin of infinite products with automorphy, but also it is very fundamental object of mathematics. We treated the additive constant, so-named Rademacher's 4-function as the additive character of the universal covering group of the modular group SL_2(Z), and obtained a new formula as follows:【numerical formula】where a, b, c, d are components of σ∈SL_2(Z), and D(h, k)=12|k|・s(h,|k|) is Dedekind sum. It seems this new formula apparently shows arithmetic and algebraic nature of itself, and it is very expected to apply to various branches of mathematics.