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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 <math>\mathbb{Q}</math> and a prime ''p''. It returns the exponent ''f''''p'' of ''p'' in the [[conductor of an elliptic curve|conductor]] of ''E'', the type of reduction at ''p'', the local index

: <math>c_p=[E(\mathbb{Q}_p):E^0(\mathbb{Q}_p)],</math>

where <math>E^0(\mathbb{Q}_p)</math> is the group of <math>\mathbb{Q}_p</math>-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 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).

Tate's algorithm was introduced by {{harvs|txt=yes|last=Tate|first=John|authorlink=John Tate|year=1975}} as an improvement of the description of the Néron model of an elliptic curve by {{harvs|txt|last=Néron|year=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
:<math>y^2+a_1xy+a_3y = x^3+a_2x^2+a_4x+a_6.\ </math>
Define:
:<math>a_{i,m}=a_i/\pi^m</math>
:<math>b_2=a_1^2+4a_2</math>
:<math>b_4=a_1a_3+2a_4^{}</math>
:<math>b_6=a_3^2+4a_6</math>
:<math>b_8=a_1^2a_6-a_1a_3a_4+4a_2a_6+a_2a_3^2-a_4^2</math>
:<math>c_4=b_2^2-24b_4</math>
:<math>c_6=-b_2^3+36b_2b_4-216b_6</math>
:<math>\Delta=-b_2^2b_8-8b_4^3-27b_6^2+9b_2b_4b_6</math>
:<math>j=c_4^3/\Delta.</math>

==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 Iν, 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
::<math>P(T) = T^3+a_{2,1}T^2+a_{4,2}T+a_{6,3}.\ </math>
:If the congruence P(T)&equiv;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 ''a''1 and π3 divides''a''4, and π4 divides ''a''6. If
::<math>Y^2+a_{3,2}Y-a_{6,4}\ </math>
: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 | zbl=0872.14041 |location=Cambridge | publisher=[[Cambridge University Press]] |edition=2nd | year=1997 | isbn=0-521-59820-6 }}
*{{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
|doi=10.2307/2007483|jstor=2007483|zbl=0493.14016
}}
* {{Citation | last1=Néron | first1=André | author1-link= André Néron | title=Modèles minimaux des variétes abèliennes sur les corps locaux et globaux | url=http://www.numdam.org/item?id=PMIHES_1964__21__5_0 | id={{MathSciNet | id = 0179172}} | year=1964 | journal=[[Publications Mathématiques de l'IHÉS]] | volume=21 | pages=5–128 | doi=10.1007/BF02684271}}
*{{citation | first=Joseph H. | last=Silverman | authorlink=Joseph H. Silverman |title= Advanced Topics in the Arithmetic of Elliptic Curves | series=[[Graduate Texts in Mathematics]] | volume=151 | publisher=[[Springer-Verlag]] | isbn=0-387-94328-5 | year=1994 | zbl=0911.14015 }}
*{{citation|chapter=Algorithm for determining the type of a singular fiber in an elliptic pencil
|last=Tate|first=John | authorlink=John Tate
|series=Lecture Notes in Mathematics
|publisher=Springer|publication-place= Berlin / Heidelberg
|issn=1617-9692
|volume=476
|editor1-last=Birch | editor1-first=B.J. | editor1-link=Bryan John Birch
| editor2-last=Kuyk | editor2-last=W.
|title=Modular Functions of One Variable IV
|doi=10.1007/BFb0097582
|year=1975
|isbn=978-3-540-07392-5
|pages=33–52
|mr=0393039
|zbl=1214.14020
}}

[[Category:Elliptic curves]]
[[Category:Number theory]]

CD48 - Revision history

en>Rjwilmsi: fix DOI