Configuration space of 8 points on the projective line and a 5-dimensional Picard modular group

Keiji Matsumoto; Masaaki Yoshida

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We find a generator $j_{1, 4}$ of the function field on the modular curve X₁(4) by means of classical theta functions θ₂ and θ₃, and estimate the normalized generator $N (j_{1, 4})$ which becomes the Thompson series of type 4C. With these modular functions we investigate some number theoretic properties.

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The purpose of this paper, which is a continuation of [2, 3], is to prove further results about arithmetic modular forms and functions. In particular we shall demonstrate here a q-expansion principle which will be useful in proving a reciprocity law for special values of arithmetic Hilbert modular functions, of which the classical results on complex multiplication are a special case. The main feature of our treatment is, perhaps, its independence of the theory of abelian varieties. ...

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