In the theory of
elliptic curves
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
, Tate's algorithm takes as input an
integral model
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with d ...
of an elliptic curve ''E'' over
, or more generally an
algebraic number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a ...
, and a prime or
prime ideal ''p''. It returns the exponent ''f''
''p'' of ''p'' in the
conductor
Conductor or conduction may refer to:
Music
* Conductor (music), a person who leads a musical ensemble, such as an orchestra.
* ''Conductor'' (album), an album by indie rock band The Comas
* Conduction, a type of structured free improvisation ...
of ''E'', the type of reduction at ''p'', the local index
:
where
is the group of
-points
whose reduction mod ''p'' is a
non-singular point
In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In ca ...
. Also, the
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
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 surface In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper morphism with connected fibers to an algebraic curve such that almost all fibers are smooth curves of genus 1. (Over an algebraically closed ...
s: 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
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his boo ...
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 as an improvement of the description of the Néron model of an elliptic curve by .
Notation
Assume that all the coefficients of the equation of the curve lie in a complete
discrete valuation ring
In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal.
This means a DVR is an integral domain ''R'' which satisfies any one of the following equivalent conditions:
# ''R' ...
''R'' with
perfect
Perfect commonly refers to:
* Perfection, completeness, excellence
* Perfect (grammar), a grammatical category in some languages
Perfect may also refer to:
Film
* Perfect (1985 film), ''Perfect'' (1985 film), a romantic drama
* Perfect (2018 f ...
residue field In mathematics, the residue field is a basic construction in commutative algebra. If ''R'' is a commutative ring and ''m'' is a maximal ideal, then the residue field is the quotient ring ''k'' = ''R''/''m'', which is a field. Frequently, ''R'' is ...
'' K'' and
maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals ...
generated by a
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...
π.
The elliptic curve is given by the equation
:
Define:
:
the
p-adic valuation
In number theory, the valuation or -adic order of an integer is the exponent of the highest power of the prime number that divides .
It is denoted \nu_p(n).
Equivalently, \nu_p(n) is the exponent to which p appears in the prime factorization of ...
of
in
, that is, exponent of
in prime factorization of
, or infinity if
:
:
:
:
:
:
:
:
:
The algorithm
*Step 1: If π does not divide Δ then the type is I
0, ''c''=1 and ''f''=0.
*Step 2: If π divides Δ but not c
4 then the type is I
v with v = v(Δ), ''c''=v, and ''f''=1.
*Step 3. Otherwise, change coordinates so that π divides ''a''
3,''a''
4,''a''
6. If π
2 does not divide ''a''
6 then the type is II, ''c''=1, and ''f''=v(Δ);
*Step 4. Otherwise, if π
3 does not divide ''b''
8 then the type is III, ''c''=2, and ''f''=v(Δ)−1;
*Step 5. Otherwise, let ''Q
1'' be the polynomial
::
.
:If π
3 does not divide ''b''
6 then the type is IV, ''c''=3 if
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 ''a''
1 and ''a''
2, π
2 divides ''a''
3 and ''a''
4, and π
3 divides ''a''
6. Let ''P'' be the polynomial
::
:If
has 3 distinct roots modulo π then the type is I
0*, ''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 π
2 divides ''a''
2 and π
3 divides ''a''
4, and π
4 divides ''a''
6. Let ''Q
2'' be the polynomial
::
.
:If
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
has a double root, change variables so the double root is 0. Then π
3 divides ''a''
3 and π
5 divides ''a''
6. If π
4 does not divide ''a''
4 then the type is III
* and ''f''=v(Δ)−7 and ''c'' = 2.
*Step 10. Otherwise if π
6 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.
Implementations
The algorithm is implemented for algebraic number fields in the
PARI/GP
PARI/GP is a computer algebra system with the main aim of facilitating number theory computations. Versions 2.1.0 and higher are distributed under the GNU General Public License. It runs on most common operating systems.
System overview
The P ...
computer algebra system, available through the function elllocalred.
References
*
*
*
*
*{{citation, chapter=Algorithm for determining the type of a singular fiber in an elliptic pencil
, last=Tate, first=John , authorlink=John Tate (mathematician)
, 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-first=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
Elliptic curves
Number theory