Elliptic fibers over non-perfect residue fields |
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Authors: | Michael Szydlo |
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Institution: | RSA Laboratories, Bedford, MA 01730, USA |
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Abstract: | Kodaira and Néron classified and described the geometry of the special fibers of the Néron model of an elliptic curve defined over a discrete valuation ring with a perfect residue field. Tate described an algorithm to determine the special fiber type by manipulating the Weierstrass equation. In the case of non-perfect residue fields, we discover new fiber types which are not on the Kodaira-Néron list. We describe these new types and extend Tate's algorithm to deal with all discrete valuation rings. Specifically, we show how to translate a Weierstrass equation into a form where the reduction type may be easily determined. Having determined the special fiber type, we construct the regular model of the curve with explicit blow-up calculations. We also provide tables that serve as a simple reference for the algorithm and which succinctly summarize the results. |
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Keywords: | Tate's algorithm Kodaira symbol Non-perfect residue field Resolution of singularities Blow-up Elliptic scheme Né ron model Reduction type Discrete valuation ring Flat regular model |
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