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Elliptic Curves








Let E be an elliptic curve.  By the modularity theorem, which was
recently proved by Breuil, Conrad, Diamond, Taylor, and Wiles there
is a two-dimensional space M of modular symbols attached to E.
Let N be the conductor of E; then M is obtained from ModularSymbols(N,2) by intersecting the kernels of T_p - a_p(E) for
sufficiently many p.






Warning: 
The computation of M can already be very resource intensive for
elliptic curves for which Conductor(E) is on the order of 5000.  
For example, the seemingly harmless expression ModularSymbols(EllipticCurve([0,6])) would bring my 
computer to its knees.



ModularSymbols(E) : CurveEll -> ModSym


ModularSymbols(E, sign) : CurveEll, RngIntElt -> ModSym



The space M of modular symbols associated to the elliptic curve E.





Example ModSym_BSD389A (H94E27)

We use the elliptic curve functions to numerically compute the Birch
and Swinnerton-Dyer conjectural order of the Shafarevich-Tate group 
of the elliptic curve 389A, which is the curve
of rank 2 with smallest conductor.  
The Birch and Swinnerton-Dyer conjecture asserts that
(L^((r))(E, 1) /r!)= 
(prod c_p  .( Sha) .( Reg)
/|E(Q)_( tor)|^2),
where r is the order of vanishing of L(E, s) at s=1.







> E := EllipticCurve(CremonaDatabase(),"389A");
> M := ModularSymbols(E);
> M;
Modular symbols space of level 389, weight 2, and dimension 2
> LRatio(M,1);  
0    


Next we compute the analytic rank and the leading
coefficient of the L-series at s=1.  (If your 
computer is very slow, use a number smaller than 300 below.)





> L1, r := LSeriesLeadingCoefficient(M,1,300);
> L1;
0.7593165002922467906576260031        
> r;        // The analytic rank is 2.
2            


Finally we check that the rank conjecture is true in this case, and
compute the conjectural order of the Shafarevich-Tate group.





> Rank(E);  // The algebraic rank is 2.
2            
> Omega := RealVolume(M,300); Omega;
4.980435433609741580582713757
> Reg := Regulator(E); Reg;
0.1524601779431437875
> #TorsionSubgroup(E);
1
> TamagawaNumber(E,389);
1
> TamagawaNumber(M,389);        // entirely different algorithm
1
> Sha := L1/(Omega*Reg); Sha;
0.9999979295234896211






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