Special Values of L-functions

Let M be an irreducible space of cuspidal modular symbols defined over Q, irreducible in the sense that M corresponds to a single Galois-conjugacy class of cuspidal newforms. Such an M can be computed using NewformDecomposition. Let f⁽¹⁾, ..., f^(d) be the (Gal)(/line(Q)/Q)-conjugate newforms that correspond to M, and write f^(d) = ∑_n=1^∞ a_n^(d) qⁿ. By a theorem of Hecke, the Dirichlet series L(f⁽ⁱ⁾, s) = ∑_n=1^∞ (a_n⁽ⁱ⁾ /n^s) extends (uniquely) to a holomorphic function on the whole complex plane. Of particular interest is the special value L(M, j) = L(f⁽¹⁾, j) ... L(f^(d), j), for any j∈{1, 2, ..., k - 1}.

In this section we describe how to approximate the complex numbers L(M, j) in Magma. If you are interested in computing individual special values L(f⁽ⁱ⁾, j), then you should use the modular forms package instead of the modular symbols package for this.

The variable prec below refers to the number of terms of the q-expansion of each f⁽ⁱ⁾ that are used in the computation, and not to the number of decimals of the answer that are correct. Thus, for example, to get a heuristic idea of the quality of an answer, you can increase prec, make another call to LSeries, and observe the difference between the two answers. If the difference is "small", then the approximation is probably "good".

LSeries(M, j, prec) : ModSym, RngIntElt, RngIntElt -> FldPrElt

The special value L(M, j), where j is an integer that lies in the critical strip, so 1 ≤j ≤k - 1 with k the weight of M. Here M is a space of modular symbols with sign 0, and prec is a positive integer which specifies the numbers of terms of q-expansions to use in the computation.

LSeriesLeadingCoefficient(M, j, prec) : ModSym, RngIntElt, RngIntElt -> FldPrElt, RngIntElt

The leading coefficient of Taylor expansion about the critical integer j and order of vanishing of L(M, s) at s=1. Thus if the series expansion of L(M, s) about s=1 is L(M, s) = a_r(s - 1)^r + a_{r + 1}(s - 1)^{r + 1} + a_{r + 2}(s - 1)^{r + 2} + ..., then the leading coefficient of L(M, s) is a_r and the order of vanishing is r.

RealVolume(M, prec) : ModSym, RngIntElt -> FldPrElt

The volume of A_M(R), which is defined as follows. Let S⊂C[[q]] be the space of cusp forms associated to M. Choose a basis f₁, ..., f_d for the free Z-module S∩Z[[q]]; one can prove that f₁, ..., f_d is also a basis for S. There is a period map Φ from integral cuspidal modular symbols H to C^d that sends a modular symbol x∈H to the d-tuple of integrals (< f₁, x >, ..., < f_d, x >)∈C^d. The cokernel of Φ is isomorphic to A_M(C). Moreover, the standard measure on the Euclidean space C^d induces a measure on A_M(R). It is with respect to this measure that we compute the volume. For more details, see Section 3.12.16 of [Ste00].

MinusVolume(M, prec) : ModSym, RngIntElt -> FldPrElt

The volume of the subgroup of A_M(C) on which complex conjugation acts as -1.

LRatio(M, j : parameters) : ModSym, RngIntElt -> FldRatElt

    Bound: RngIntElt                    Default: -1

The rational number L(A, j).(j - 1)! /(2π)^{j - 1}.Ω, where j is a "critical integer", so 1≤j ≤k - 1, and Ω is RealVolume(M) when j is odd and MinusVolume(M) when j is even. If the optional parameter Bound is set, then LRatio is only a divisibility upper bound on the above rational number. If Sign(M) is not 0, then LRatio(M,j) is only correct up to a power of 2.

LRatioOddPart(M, j) : ModSym, RngIntElt -> FldRatElt

The odd part of the rational number LRatio(M,j). Hopefully, computing LRatioOddPart(M,j) takes less time than finding the odd part of LRatio(M,j).

> M := ModularSymbols(11,2);
> C := CuspidalSubspace(M);
> LSeries(C,1,100);
0.2538418608559106843377589233

> A := ModularSymbols("65B"); A;  // <--> dimension two abelian variety
Modular symbols space of level 65, weight 2, and dimension 4 
> LSeries(A,1,100);
0.9122515886981898410935140211 +  0.E-29*i

    Bound: RngIntElt                    Default: -1

    Bound: RngIntElt                    Default: -1