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