Lesson plan
Tate's algorithm
In the theory of elliptic curves, Tate's algorithm takes as input an integral model of an elliptic curve E over \mathbb{Q} , or more generally an algebraic number field, and a prime or prime ideal p. It returns the exponent f p of p in the conductor of E, the type of reduction at p, the local index : c_p=[E(\mathbb{Q}_p):E^0(\mathbb{Q}_p)], where E^0(\mathbb{Q}_p) is the group of \mathbb{Q}_p -points whose reduction mod p is a non-singular point. Also, the algorithm determines whether or not the given integral model is minimal at p, and, if not, returns an integral model with integral coefficients for which the valuation at p of the discriminant is minimal. Tate's algorithm also gives the structure of the singular fibers given by the Kodaira symbol or Néron symbol, for which, see elliptic surface: in turn this determines the exponent f p of the conductor E. Tate's algorithm can be greatly simplified if the characteristic of the residue class field is not 2 or 3; in this case the type and c and f can be read off from the valuations of j and Δ (defined below). Tate's algorithm was introduced by as an improvement of the description of the Néron model of an elliptic curve by .
Article summary
In the theory of elliptic curves, Tate's algorithm takes as input an integral model of an elliptic curve E over \mathbb{Q} , or more generally an algebraic number field, and a prime or prime ideal p. It returns the exponent f p of p in the conductor of E, the type of reduction at p, the local index : c_p=[E(\mathbb{Q}_p):E^0(\mathbb{Q}_p)], where E^0(\mathbb{Q}_p) is the group of \mathbb{Q}_p -points whose reduction mod p is a non-singular point. Also, the algorithm determines whether or not the given integral model is minimal at p, and, if not, returns an integral model with integral coefficients for which the valuation at p of the discriminant is minimal. Tate's algorithm also gives the structure of the singular fibers given by the Kodaira symbol or Néron symbol, for which, see elliptic surface: in turn this determines the exponent f p of the conductor E. Tate's algorithm can be greatly simplified if the characteristic of the residue class field is not 2 or 3; in this case the type and c and f can be read off from the valuations of j and Δ (defined below). Tate's algorithm was introduced by as an improvement of the description of the Néron model of an elliptic curve by .
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
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