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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 Tp - ap(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, sign) : CrvEll, RngIntElt -> ModSym
The space M of modular symbols associated to the elliptic curve E.
Database: BoolElt Default: true
An elliptic curve over the rational numbers that lies in the
isogeny class of elliptic curves associated to M.
By default, the Cremona database is used. To compute the curve
from scratch, set the optional parameter Database to false.
(Note however that this is not optimized for large level.)
CoefficientPrecision: RngIntElt Default: 3
SeriesPrecision: RngIntElt Default: 5
QuadraticTwist: RngIntElt Default: 1
ProperScaling: BoolElt Default: true
Given an elliptic curve over the rationals and a prime of multiplicative
or good ordinary reduction, compute the p-adic L-series.
For technical reasons involving caching, the curve is required to be
given by a (global) minimal model.
This computation can be rather time-consuming for large conductors
and/or primes.
The algorithm needs to compute the modular symbols (taking N3 time or so),
and typically needs to evaluate 2(p - 1)pn of them, where n is the
desired coefficient precision. Also, the desired coefficient precision
does not always apply to every term in the series.
The QuadraticTwist vararg computes the p-adic L-series for a twist
by a fundamental discriminant D. This requires |D| as many modular symbols
to be computed, but in most cases is much faster than computing the modular
symbols for the twisted curve. The twisting discriminant is required to be
coprime to the conductor of the curve.
The ProperScaling vararg indicates whether to ensure that the scaling
of the p-adic L-series is correct. For many applications, only the
valuation of the coefficients is needed. The computation of the proper scaling
induces a bit of extra overhead, but is still typically faster than computing
the modular symbols for examples of interest.
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!)=
(∏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 := Periods(M,300)[2][1] * 2; 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
> f := pAdicLSeries(E,3); _<T> := Parent(f); f;
O(3^11) + O(3^3)*T - (1 + O(3^3))*T^2 + (2 + O(3^2))*T^3
- (2 + O(3^2))*T^4 + (1 + O(3^2))*T^5 + O(T^6)
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