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The characteristic polynomial of a monodromy transformation attached to a family of curves. (English) Zbl 0801.14006

Let \(K\) be a complete field with respect to a discrete valuation \(\nu\). Let \(k\) be the residue field and suppose that it is algebraically closed and \(p \geq 0\) the residue characteristic. Let \(X/K\) be a curve of genus \(g\) having a \(K\)-rational point. In this work the author introduces a polynomial \(f_{X/K} (x)\) of degree \(2g\) attached to the special fiber of a regular model of \(X/K\) and obtains a homological interpretation of a factor \(f^{(p)}_{X/K}(x)\) of the polynomial \(f_{X/K} (x)\), so as a description of the behavior of the polygonal \(f_{X/K} (x)\) under base changes. Finally he describes the characteristic polynomial of an automorphism of a curve acting on the first homological group of the curve.

MSC:

14H10 Families, moduli of curves (algebraic)
14F25 Classical real and complex (co)homology in algebraic geometry
14H70 Relationships between algebraic curves and integrable systems
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