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Elliptic curves with CM unveil a new phenomenon in the theory of large algebraic fields. Rather than drawing a line between and or and they give an example where the line goes beween and and another one where the line goes between and .
We propose a definition of sign of imaginary quadratic fields. We give an example of such
functions, and use it to define new invariants that are roots of the classical
Ramachandra invariants. Also we introduce signed ordinary distributions and compute their
signed cohomology by using Anderson's theory of double complex.
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