Tate's algorithm

In the theory of elliptic curves, Tate's algorithm, described by harvs|txt=yes|last=Tate|first=John|authorlink=John Tate|year=1975, 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 $mathbb\{Q\}\_p$-pointswhose 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 $p$.

The type of reduction is given by the Kodaira 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).

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::$a\_\{i,m\}=a\_i/pi^m$:$b\_2=a\_1^2+4a\_2$:$b\_4=a\_1a\_3+2a\_4^\{\}$:$b\_6=a\_3^2+4a\_6$:$b\_8=a\_1^2a\_6-a\_1a\_3a\_4+4a\_2a\_6+a\_2a\_3^2-a\_4^2$:$c\_4=b\_2^2-24b\_4$:$c\_6=-b\_2^3+36b\_2b\_4-216b\_6$:$Delta=-b\_2^2b\_8-8b\_4^3-27b\_6^2+9b\_2b\_4b\_6$:$j=c\_4^3/Delta$

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 "a"₃,"a"₄,"a"₆. If π does not divide "b"₂ then the type is I_ν, with ν =v(Δ), and "f"=1.
*Step 3. Otherwise, if π² does not divide "a"₆ then the type is II, "c"=1, and "f"=v(Δ);
*Step 4. Otherwise, if π³ does not divide "b"₈ then the type is III, "c"=2, and "f"=v(Δ)−1;
*Step 5. Otherwise, if π³ does not divide "b"₆ then the type is IV, "c"=3 or 1, and "f"=v(Δ)−2.
*Step 6. Otherwise, change coordinates so that π divides "a"₁ and "a"₂, π² divides "a"₃ and "a"₄, and π³ divides "a"₆. 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 "a"₁ and π³ divides"a"₄, and π⁴ divides "a"₆. 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 "a"₃ and π⁵ divides "a"₆. If π⁴ does not divide "a"₄ then the type is III^* and "f"=v(Δ)−7 and "c" = 2.
*Step 10. Otherwise if π⁶ does not divide "a"₆ 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 π^"n" and go back to step 1.

References

*citation

last = Cremona

first = John

title = Algorithms for modular elliptic curves

accessdate = 2007-12-20

url = http://www.warwick.ac.uk/~masgaj/book/fulltext/index.html

*citation|title=An Algorithm for Finding a Minimal Weierstrass Equation for an Elliptic Curve
first= Michael |last=Laska

journal=Mathematics of Computation|volume= 38|issue= 157|year= 1982|pages= 257-260

url= http://links.jstor.org/sici?sici=0025-5718%28198201%2938%3A157%3C257%3AAAFFAM%3E2.0.CO%3B2-R

*citation|chapter=Algorithm for determining the type of a singular fiber in an elliptic pencil
last=Tate|first=John
series=Lecture Notes in Mathematics
publisher=Springer|publication-place= Berlin / Heidelberg
ISSN= 1617-9692
volume= 476
title=Modular Functions of One Variable IV
year=1975
ISBN=978-3-540-07392-5
DOI=10.1007/BFb0097582
pages=33-52
id=MR|0393039