# Download Seminar on Complex Multiplication by A. Borel, S. Chowla, C. S. Herz, K. Iwasawa, J. P. Serre PDF

By A. Borel, S. Chowla, C. S. Herz, K. Iwasawa, J. P. Serre

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Hence, CO L(s; ~ ) ~ ~ l ~-~' V~ % with rational integers a ~ ~65 ~o" ~ O. R(s) > l, Suppose now that L(1; ~ ) = O for some Then the left hand side of the above would be holomorphic on the entire s-plane and, by a theorem of Landau on the convergence of Dirichlet series with positive coefficients, the right hand side of the above must converge for all s. Since a R are integers, it would then follow that a y = except for a finite number -- a contradiction. )-V log L(s; ~ ) ~ z, Z=l C ~ (2)v s ,-, z, ~ (~) NK/Q(p)'s S >I.

N THEOREM 1. (a) Fn(t , j) - ~ - (t - js(Z)) is a polynomial of t and s =I j = j(z) ~rith integral rational coefficients, (b) i_~fn is not a square, the highest coefficient of j in Fn(J, j) is + l, (c) Fn(t , J) is an irreducible polynomial of t over the field C=(j), (d) Fn(t , j) = Fn(J, t), n > 1. Proof. , jN(z)) of Jl' "''' JN is, as noticed above, invariant under G and is obviously a holomorphic function of z in E. q = e 2~iz. Then, by II w To see the behavior of ~ at co, put j(z) = q'l(l + A(q)) where A(q) is a power series of q with integral rational coefficients and A(O) = O.

And ~(P,,) = o, ~(P,) - ~(P) - 1. More generally, if we put M' = M6~ P', M" - M g ~ P " for any subset M of P, then ~(M") = O, l(M') = &(M), whenever one of ~(M') and ~ ( M ) exists. Now, let H be an ideal group of K defined mod. m and let k be any class (coset) of I m p . - HI. This is the theorem of arithmetic progression for the field K, which generalizes the well-known theorem of Dirichlet for K = ~. An outline of the proof of E. Hecke for (I) is as follows (of. GSttin~er Nachrichten, 1917) : Let L(s;X) ~ .