By Nagell T.

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28. Let K be a congruence subgroup of Sn , and χ a congruence character of K. 60) ← − − χ ) = C. where for n = 1, we set M ( K , ← k Proof. One can assume that n > 1. 58). Then we have M Zλ = A Z +B 0 0 iλ det(CZλ + D) = det(C Z + D ), C Z +D 0 0 1 and −1 =M Z λ, −(M ). χ (M) = ← χ It follows that (F|Φ|k M )(Z ) = det(C Z + D )−k lim F(M Z λ →+∞ −k λ) = lim det(CZλ + D) F(M Zλ ) = (F|k M|Φ)(Z ). 61) 30 1 Modular Forms ← − In particular, if M ∈ K , then we have −(M )F|Φ. 61) that the function (F|Φ|k M )(Z ) is with ε > 0.

34), the heights of the points M Z and g(u) satisfy the relation h(M Z ) = | det(CZ + D)|−2 h(Z) = v ∆(g, u) 2 = h(g(u)). It follows that the values of the height of points of the orbit Λ (u) are among the values of the heights of points of the orbit Γ2 Z , where u(Z) = u. 15, we conclude that each orbit of Λ on L contains points u of maximal height and the points can be characterized by inequalities ∆(g, u) ≥ 1 for every g ∈ Λ . 16, in order to construct a fundamental domain for Λ on L, first of all, we shall choose on the Λ -orbit of a point u ∈ L a point u = (z , v ) of maximal hight.

44). Proof. By analogy with the symplectic case, let us call the positive real number v2 = h(u) the height of a point u = (z, v) ∈ L. 34), the heights of the points M Z and g(u) satisfy the relation h(M Z ) = | det(CZ + D)|−2 h(Z) = v ∆(g, u) 2 = h(g(u)). It follows that the values of the height of points of the orbit Λ (u) are among the values of the heights of points of the orbit Γ2 Z , where u(Z) = u. 15, we conclude that each orbit of Λ on L contains points u of maximal height and the points can be characterized by inequalities ∆(g, u) ≥ 1 for every g ∈ Λ .

### Introduction to number theory by Nagell T.

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