Complex Conjugation

Let \(A\) be an étale algebra over \({{\Bbb Q}}\) with components \(K_1\times\cdots\times K_n\). We say that \(A\) is a CM-étale algebra if every component \(K_i\) is a CM-field, that is, \(K_i\) has an involution that acts as applying complex conjugation after applying any homomorphism to the complex numbers. If \(A\) is a CM-étale algebra, then it has an involution with the same property. For this reason, we call this involution complex conjugation and denote it as \(\overline{\cdot}\).

Given an element of \(A\), an order or a fractional ideal in \(A\), we say that it is conjugate stable if it equals its complex conjugate. An element \(x\) of \(A\) is called totally real if \(x=\overline{x}\) and totally imaginary if \(x=-\overline{x}\). A totally real element \(a\) is called totally positive (resp. totally negative) if \(\varphi(a) > 0\) (resp. \(\varphi(a)<0\)) for every homomorphism \(\varphi: A \to {\Bbb C}\).

HasComplexConjugate(A): AlgEtQ -> BoolElt

Returns if the algebra \(A\) is the product of CM fields.

ComplexConjugate(x): AlgEtQElt -> AlgEtQElt

If the algebra \(A\) of the element \(x\) is a product of CM fields, it returns the complex conjugate of the argument.

IsConjugateStable(O): AlgEtQOrd -> BoolElt, AlgEtQOrd

Given an order \(O\) in a CM-étale algebra, it returns whether \(O\) is conjugate stable and the complex conjugate.

ComplexConjugate(O): AlgEtQOrd -> AlgEtQOrd

Given an order \(O\) in a CM-étale algebra, it returns the complex conjugate of \(O\).

IsConjugateStable(I): AlgEtQIdl -> BoolElt, AlgEtQIdl

Given a fractional ideal \(I\) in a CM-étale algebra, it returns whether \(I\) is conjugate stable and the complex conjugate. Note: if the order of \(I\) is not conjugate stable, then the second output will be defined over the complex conjugate of the order.

ComplexConjugate(I): AlgEtQIdl -> AlgEtQIdl

If \(A\) is a product of CM fields, it returns the complex conjugate of the fractional ideal \(I\). Note: if the order of \(I\) is not conjugate stable, then the output will be defined over the complex conjugate of the order.