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 fp 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 surfaces: in turn this determines the exponent fp 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 John Tate (1975) as an improvement of the description of the Néron model of an elliptic curve by Néron (1964).

Notation

Assume that all the coefficients of the equation of the curve lie in a complete discrete valuation ring R with perfect residue field and maximal ideal generated by a prime π. The elliptic curve is given by the equation

y^2+a_1xy+a_3y = x^3+a_2x^2+a_4x+a_6.\

Define:

a_{i,m}=a_i/\pi^m
b_2=a_1^2+4a_2
b_4=a_1a_3+2a_4^{}
b_6=a_3^2+4a_6
b_8=a_1^2a_6-a_1a_3a_4+4a_2a_6+a_2a_3^2-a_4^2
c_4=b_2^2-24b_4
c_6=-b_2^3+36b_2b_4-216b_6
\Delta=-b_2^2b_8-8b_4^3-27b_6^2+9b_2b_4b_6
j=c_4^3/\Delta.

The algorithm

P(T) = T^3+a_{2,1}T^2+a_{4,2}T+a_{6,3}.\
If the congruence P(T)≡0 has 3 distinct roots then the type is I0*, f=v(Δ)−4, and c is 1+(number of roots of P in k).
Y^2+a_{3,2}Y-a_{6,4}\
has distinct roots, the type is IV*, f=v(Δ)−6, and c is 3 if the roots are in k, 1 otherwise.

References