Tate's algorithm

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In the theory of elliptic curves, Tate's algorithm takes as input an integral model of an elliptic curve E over ${\mathbb {Q}}$ and a prime 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 which is minimal at p.

Tate's algorithm also gives the structure of the singular fibers given by the Kodaira symbol or Néron symbol, for which, see elliptic surfaces.

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

Define:

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

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

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

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

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

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

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

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

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

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 I_ν, 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

$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).

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.
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

$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.

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 a_n by πⁿ and go back to step 1.

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 variétes abèliennes sur les corps locaux et globaux", Publications Mathématiques de l'IHÉS 21: 5–128, doi:10.1007/BF02684271, MR 0179172
Silverman, Joseph H. (1994), Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Mathematics 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.; W., Modular Functions of One Variable IV, Lecture Notes in Mathematics 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

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