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Uniformizable families of \(t\)-motives. (English) Zbl 1140.11030
In [Duke Math. J. 53, 457–502 (1986; Zbl 0679.14001)], G. W. Anderson showed that the following are equivalent:
(1) an Abelian \(t\)-module E is uniformisable, which roughly means that it permits an analytic description as a quotient \(V/\Lambda\) of a finite dimensional \(\mathbb C_\infty\)-vector space \(V\) by a free and discrete sub-\(\mathbb F_q[t]\)-module \(\Lambda\),
(2) the associated \(t\)-motive \(M\) is analytically trivial, which roughly means that it admits a \(\tau\)-invariant basis after extending scalars from \(\mathbb C_{\infty[t]}\) to the Tate algebra \(\mathbb C\langle t\rangle\)
(the reader who is unfamiliar with these notions is strongly encouraged to consult Anderson (loc. cit.) first).
The paper under review studies how these notions behave in families.
The locus of uniformisable \(t\)-modules in an algebraic family is usually not algebraic: it typically depends on inequalities between valuations of coordinates. (The reader who is unfamiliar with this phenomenon is encouraged to read section 7 of the paper under review first, it contains a detailed treatment of an example due to Pink.) It is therefore most natural to study uniformisability in families over a rigid analytic base, and this is exactly what the authors do. After carefully working out the notions of a rigid analytic family of \(t\)-motives (and of \(t\)-modules), and of analytic triviality (and uniformisability) of such families.
The main result (Theorem 5.5) states that the locus of uniformisability in a rigid analytic family is Berkovich open (i.e., the set of analytic points over which the given family is uniformisable is open in the topological space of all analytic points of the base.)
The exposition is excellent and the authors have done considerable effort to recall the necessary notions and results from rigid analytic geometry.

MSC:
11G09 Drinfel’d modules; higher-dimensional motives, etc.
14G22 Rigid analytic geometry
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