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 surfaces: 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 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 K and maximal ideal generated by a prime π. The elliptic curve is given by the equation

$y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}.$

Define:

$v(\Delta )=$ the p-adic valuation of $\pi$ in $\Delta$, that is, exponent of $\pi$ in prime factorization of $\Delta$, or infinity if $\Delta =0$

$a_{i,m}=a_{i}/\pi ^{m}$

$b_{2}=a_{1}^{2}+4a_{2}$

$b_{4}=a_{1}a_{3}+2a_{4}^{}$

$b_{6}=a_{3}^{2}+4a_{6}$

$b_{8}=a_{1}^{2}a_{6}-a_{1}a_{3}a_{4}+4a_{2}a_{6}+a_{2}a_{3}^{2}-a_{4}^{2}$

$c_{4}=b_{2}^{2}-24b_{4}$

$c_{6}=-b_{2}^{3}+36b_{2}b_{4}-216b_{6}$

$\Delta =-b_{2}^{2}b_{8}-8b_{4}^{3}-27b_{6}^{2}+9b_{2}b_{4}b_{6}$

$j=c_{4}^{3}/\Delta .$

The algorithm

• Step 1: If π does not divide Δ then the type is I₀, c=1 and f=0.
• Step 2: If π divides Δ but not c₄ then the type is I_v with v = v(Δ), c=v, and f=1.
• Step 3. Otherwise, change coordinates so that π divides a₃,a₄,a₆. 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, let Q₁ be the polynomial

$Q_{1}(Y)=Y^{2}+a_{3,1}Y-a_{6,2}.$.

If π³ does not divide b₆ then the type is IV, c=3 if $Q_{1}(Y)$ has two roots in K and 1 if it has two roots outside of K, 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

$P(T)=T^{3}+a_{2,1}T^{2}+a_{4,2}T+a_{6,3}.$

If $P(T)$ has 3 distinct roots modulo π 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 I_ν^* for some ν>0, f=v(Δ)−4−ν, c=2 or 4: there is a "sub-algorithm" for dealing with this case.
• Step 8. If P has a triple root, change variables so the triple root is 0, so that π² divides a₂ and π³ divides a₄, and π⁴ divides a₆. Let Q₂ be the polynomial

$Q_{2}(Y)=Y^{2}+a_{3,2}Y-a_{6,4}.$.

If $Q_{2}(Y)$ has two distinct roots modulo π then the type is IV^*, f=v(Δ)−6, and c is 3 if the roots are in K, 1 otherwise.

• Step 9. If $Q_{2}(Y)$ 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 a_n by πⁿ and go back to step 1.

Implementations

The algorithm is implemented for algebraic number fields in the PARI/GP computer algebra system, available through the function elllocalred.

References

• Cremona, John (1997), Algorithms for modular elliptic curves (2nd ed.), Cambridge: Cambridge University Press, ISBN 0-521-59820-6, Zbl 0872.14041, retrieved 2007-12-20
• Laska, Michael (1982), "An Algorithm for Finding a Minimal Weierstrass Equation for an Elliptic Curve", Mathematics of Computation, 38 (157): 257–260, doi:10.2307/2007483, JSTOR 2007483, Zbl 0493.14016
• Néron, André (1964), "Modèles minimaux des variétés abèliennes sur les corps locaux et globaux", Publications Mathématiques de l'IHÉS (in French), 21: 5–128, doi:10.1007/BF02684271, MR 0179172, Zbl 0132.41403
• Silverman, Joseph H. (1994), Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, ISBN 0-387-94328-5, Zbl 0911.14015
• Tate, John (1975), "Algorithm for determining the type of a singular fiber in an elliptic pencil", in Birch, B.J.; Kuyk, W. (eds.), Modular Functions of One Variable IV, Lecture Notes in Mathematics, vol. 476, Berlin / Heidelberg: Springer, pp. 33–52, doi:10.1007/BFb0097582, ISBN 978-3-540-07392-5, ISSN 1617-9692, MR 0393039, Zbl 1214.14020