[Next] [Prev] [Right] [Left] [Up] [Index] [Root] 
Kodaira Symbols


Kodaira Symbols






Kodaira symbols have their own type KodSym. Apart from the two functions
that determine symbols for elliptic curves, there is a special creation
function and a comparison operator, to test Kodaira symbols.



KodairaSymbol(E, p) : CurveEll -> MonStgElt



Given an elliptic curve E defined over Q and a prime number p,
this function returns the reduction type
of E modulo p in the form of a Kodaira symbol.



KodairaSymbols(E) : CurveEll -> MonStgElt



Given an elliptic curve E defined over Q,
this function returns the reduction types
of E modulo the bad primes in the form of a sequence of
Kodaira symbols.



KodairaSymbol(s) : MonStgElt -> KodSym



Given a string s,  return a Kodaira symbol it represents.
The values of s that are allowed are:
"I0", "I1", "I2", ..., "In", "II", "III", "IV", and
"I0*", "I1*", "I2*", ..., "In*", "II*", "III*", "IV*".
The dots stand for "Ik" with k a positive integer.
The `generic' type "In" is provided to be able to match
types "I"n for any integer n>0 (and similarly for "In*").
This creation function is primarily useful for comparing values returned by
the functions determining the Kodaira symbol for a curve. 



h eq k : KodSym, KodSym -> BoolElt



Given two Kodaira symbols h and k, this function returns true if and only
if either both are identical, or one is generic (of the form "In", or "In*")
and the other is specific of the same type: "In" will compare
equal with any of "I1", "I2", "I3", etc., and "In*" will compare equal with
any of "I1*", "I2*", "I3*", etc.  Note however that "In" and "I3" are
different from the point of view of set creation.



ReductionType(E, p) : CurveEll, RngIntElt -> MonStgElt



Returns a string describing the reduction type: the possibilities are
"Good", "Additive", "Split multiplicative" or "Unsplit multiplicative".
These correspond to the type of singularity (if any) on the reduced curve.
This function is necessary as the Kodaira symbols do not distinguish
between split and unsplit multiplicative reduction. 





Example Elcu_Kodaira (H53E12)

We search for curves with a particular reduction type `I0*' in
a family of curves.





> for n := 2 to 100 do                              
>    E := EllipticCurve([n,0]);
>    for p in BadPrimes(E) do
>       if KodairaSymbol(E, p) eq KodairaSymbol("I0*") then
>          print p, n;
>       end if;
>    end for;
> end for;                          
3 9
3 18
5 25
3 36
3 45
7 49
5 50
3 63
3 72
5 75
3 90
7 98
3 99
5 100





[Next] [Prev] [Right] [Left] [Up] [Index] [Root]