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Arithmetic








The standard arithmetic operations of addition, subtraction, and 
scalar multiplication are defined for elements of supersingular modules.  
The modules themselves support operations of addition and intersection.



P + Q : ModSSElt, ModSSElt -> ModSSElt



P - Q : ModSSElt, ModSSElt -> ModSSElt



a * P : RngElt, ModSSElt -> ModSSElt




The sum, difference, or scalar product of supersingular module elements. 





M1 + M2 : ModSS, ModSS -> ModSS



The submodule generated by all sums of elements in M_1 and M_2.



M1 meet M2 : ModSS, ModSS -> ModSS



The intersection of M_1 and M_2.





Example ModSS_Arithmetic (H92E7)

We begin with the elementary construction of elements and their arithmetic.





> M := SupersingularModule(11);
> P := M.1; P;
(1, 1)
> Q := M.2; Q;
(0, 0)
> P + Q;
(1, 1) + (0, 0)
> P - Q;
(1, 1) - (0, 0)
> 3*P;
3*(1, 1)


Then we follow with some arithmetic on submodules.





> E := EisensteinSubspace(M);
> S := CuspidalSubspace(M);
> V := E + S;
> V;
Supersingular module associated to X_0(1)/GF(11) of dimension 2
> Basis(V);
[
    (1, 1) + 4*(0, 0),
    5*(0, 0)
]


Because the modules are all modules over the integers, the index of E + S 
in M is of interest.  Upon converting each to an RSpace, we find 
that the index is Z/5Z.  Fans of Mazur's Modular Curves and the 
Eisenstein Ideal [Maz77] will recognize that 5 as the 
numerator of (11 - 1)/12=5/6.





> RSpace(M)/RSpace(V);
Full Quotient RSpace of degree 1 over Integer Ring
Column moduli:
[ 5 ]


The intersection of E and S is the zero module.





> W := E meet S; W;
Supersingular module associated to X_0(1)/GF(11) of dimension 0
> Basis(W);
[]





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