Tate's algorithm
Encyclopedia
In the theory of elliptic curves, Tate's algorithm, described by , takes as input an integral model of an elliptic curve 
over 
and a prime 
. It returns the exponent 
of 
in the conductor of 
, the type of reduction at 
, the local index
where
is the group of 
-points
whose reduction mod
is a non-singular point. Also, the algorithm
determines whether or not the given integral model is minimal at
, and, if not, returns an integral model which is minimal at 
.
The type of reduction is given by the Kodaira symbol, for which, see elliptic surface
s.
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).
The elliptic curve is given by the equation
Define:
over and a prime . It returns the exponent of in the conductor of , the type of reduction at , the local index
where
is the group of -pointswhose reduction mod
is a non-singular point. Also, the algorithmAlgorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
determines whether or not the given integral model is minimal at
, and, if not, returns an integral model which is minimal at .The type of reduction is given by the Kodaira symbol, for which, see elliptic surface
Elliptic surface
In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper connected morphism to an algebraic curve, almost all of whose fibers are elliptic curves....
s.
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).
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
Define:
Tate's algorithm
- Step 1: If π does not divide Δ then the type is I0, f=0, c=1.
- Step 2. Otherwise, change coordinates so that π divides a3,a4,a6. If π does not divide b2 then the type is Iν, with ν =v(Δ), and f=1.
- Step 3. Otherwise, if π2 does not divide a6 then the type is II, c=1, and f=v(Δ);
- Step 4. Otherwise, if π3 does not divide b8 then the type is III, c=2, and f=v(Δ)−1;
- Step 5. Otherwise, if π3 does not divide b6 then the type is IV, c=3 or 1, and f=v(Δ)−2.
- Step 6. Otherwise, change coordinates so that π divides a1 and a2, π2 divides a3 and a4, and π3 divides a6. Let P be the polynomial
-
- 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).
- Step 7. If P has one single and one double root, then the type is Iν* for some ν>0, f=v(Δ)−4−ν, c=2 or 4.
- Step 8. If P has a triple root, change variables so the triple root is 0, so that π2 divides a1 and π3 dividesa4, and π4 divides a6. If
- has distinct roots, the type is IV*, f=v(Δ)−6, and c is 3 if the roots are in k, 1 otherwise.
- Step 9. The equation above has a double root. Change variables so the double root is 0. Then π3 divides a3 and π5 divides a6. If π4 does not divide a4 then the type is III* and f=v(Δ)−7 and c = 2.
- Step 10. Otherwise if π6 does not divide a6 then the type is II* and f=v(Δ)−8 and c = 1.
- Step 11. Otherwise the equation is not minimal. Divide each an by πn and go back to step 1.
