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Trace on \(\mathbb{C}_p\). (English) Zbl 0965.11049

For a prime number \(p\), let \({\mathbb Q} _p\) be the field of \(p\)-adic numbers and let \({\mathbb C} _p\) be the completion of a fixed algebraic closure \(\overline{{\mathbb Q}} _p\) of \({\mathbb Q} _p\). The authors construct for elements \(T\) of \({\mathbb C} _p\) satisfying certain diophantine condition a power series \(F(T,Z)\) with coefficients in \({\mathbb Q} _p\) and prove that \(F(T,Z)=F(U,Z)\) if and only if \(U\) and \(T\) are conjugate over \({\mathbb Q} _p\). The reason for introducing this series \(F(T,Z)\) is that when it is expanded around zero, the coefficient of \(Z\) in \(F(T,Z)\) is viewed as the trace of \(T\).
Next, the series \(F(T,Z)\) is studied as a rigid analytic function and it is obtained that for any fixed \(T \in {\mathbb C} _p\), \(F(T,Z)\) is a rigid analytic function defined on \({\mathbb C} _p \setminus C(1/T)\) where \(C(1/T)\) is the set of conjugates of \(1/T\). As a consequence of this study, it is derived the main result of this paper which asserts that if \(\{T _\alpha\} _\alpha\) is a family of elements of \({\mathbb C} _p\) which are non-conjugate, transcendental over \({\mathbb Q} _p\) and satisfy the diophantine condition mentioned above, the functions \(\{F(T_\alpha, Z)\}_\alpha\) are algebraically independent over \({\mathbb C} _p (Z)\). In particular, if \(T\) is an element of \({\mathbb C} _p\) satisfying the diophantine condition, then \(F(T,Z)\) is transcendental over \({\mathbb C} _p (Z)\) if and only if \(T\) is transcendental over \({\mathbb Q} _p\).
The paper runs as follows. The first two sections present some definitions and general results. In the next section, it is proved that any \(T \in {\mathbb C} _p\) produces a measure \(\mu _T\) on \({\mathbb C} _p\) which is related to the Haar measure on the group \(G = \text{ Gal}(\overline{{\mathbb Q}} _p/{\mathbb Q} _p)\). Also, the Haar measure on \(G\) induces a \(p\)-adic measure \(\pi _T\) on \({\mathbb C} _p\). In Section 4 it is studied integration with respect to the \(p\)-adic Haar measure \(\pi _T\). In Sections 5 and 6 it is defined the trace of an element and the “trace function” \(F(T,Z)\). In Section 7 the authors state the main result. In Section 8 it is studied the convergence of the sequence of measures \(\{\mu _{\alpha _n}\}_n\) for a sequence of elements \(\{\alpha _n\}_n\) of \(\overline{{\mathbb Q}} _p\) that converges to \(T\). The measure \(\mu _T\) is then used in the study of a metric invariant which is defined in Section 9. Finally, in the last section they complete the proof of the main result.

MSC:

11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
12F20 Transcendental field extensions
Full Text: DOI

References:

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