By Martin Eichler; Translated by George Striker

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T In the case n = 1 an additional assumption on the behavior of the functions as T = t approaches the real axis is needed; see Chapter IV, $4. For n > 1 these modular functions do not fall within the scope of this book. x) + 7. PROOFOF THE RECIPROCITY FORMULA The proof of (12) is carried out under the additional assumption that T is purely imaginary. As an analytic function in the complex variables t i , , x i , y , is being dealt with, the functional equation continues to hold under analytic continuation.

7 The discussion also yields the fact that those substitutions M,which transform the fundamental domain into a neighboring domain generate the group r'. This clearly holds true for all groups which have fundamental domains. On the other hand, a system of generators of a group need, by no means, correspond to a fundamental domain in this sense. -1 -+ 0 + 1 + 9 . - + n) 41 $1. t It is difficult to describe because of the behavior of the subgroup of the U, . $ 6. THETHETA FUNCTION We introduce a new notation before defining this function.

F a,, = n ( x - ai) (al = a) I=1 of a E K be of degree n. , an-' form a basis of K/k, and the corresponding matrix has the form which implies n n s(a) = a , = C a I , I=1 n(a) = a, = n a I . I=1 2. THETRANSITIVITY FORMULAS Let an intermediate field L be given: k c L c K. One can take traces and norms both with respect to L and to k, the difference being clearly expressed in an obvious terminology. We will prove the transitivityformulas sK/k(a)= SL/k(sK/L(a)), nK/k(a) = nL/k(nR/L(a)). , q,,, be bases of KIL and Llk.