Special Values of \(L\)-functions

Let \(M\) be an irreducible space of cuspidal modular symbols defined over \({\mathbb{Q}}\), 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 \(f^{(1)},\ldots,f^{(d)}\) be the \({\rm Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})\)-conjugate newforms that correspond to \(M\), and write \(f^{(d)} = \sum_{n=1}^{\infty} a_n^{(d)} q^n\). By a theorem of Hecke, the Dirichlet series

\[L(f^{(i)},s) = \sum_{n=1}^{\infty} {a_n^{(i)} \over n^{s}}\]

extends (uniquely) to a holomorphic function on the whole complex plane. Of particular interest is the special value

\[L(M,j) = L(f^{(1)},j) \cdots L(f^{(d)},j),\]

for any \(j\in \{1,2,\ldots,k-1\}\).

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 \(1 \leq j \leq k-1\) 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

\[L(M,s) = a_r(s-1)^r + a_{r+1}(s-1)^{r+1} + a_{r+2}(s-1)^{r+2} + \cdots,\]

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 \(A_M({\mathbb{R}})\), which is defined as follows. Let \(S\subset {\mathbb{C}}[[q]]\) be the space of cusp forms associated to \(M\). Choose a basis \(f_1,\ldots, f_d\) for the free \({\mathbb{Z}}\)-module \(S\cap {\mathbb{Z}}[[q]]\); one can prove that \(f_1,\ldots, f_d\) is also a basis for \(S\). There is a period map \(\Phi\) from integral cuspidal modular symbols \(H\) to \({\mathbb{C}}^d\) that sends a modular symbol \(x\in H\) to the \(d\)-tuple of integrals \((\langle f_1, x\rangle, \ldots, \langle f_d, x\rangle)\in {\mathbb{C}}^d\). The cokernel of \(\Phi\) is isomorphic to \(A_M({\mathbb{C}})\). Moreover, the standard measure on the Euclidean space \({\mathbb{C}}^d\) induces a measure on \(A_M({\mathbb{R}})\). 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 \(A_M({\mathbb{C}})\) on which complex conjugation acts as \(-1\).

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

Bound: RngIntElt                    Default: -1

The rational number

\[L(A,j)\cdot (j-1)! \over (2\pi)^{j-1}\cdot \Omega,\]

where \(j\) is a “critical integer”, so \(1\leq j \leq k-1\), and \(\Omega\) 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).

Example: L-Series Values of Modular Symbols (ex-75fe6d)

> 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

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Winding Elements

Let \(\bf M_k(N)\) be a space of modular symbols over \({\mathbb{Q}}\). For \(i=1,\ldots,k\), the \(i\)th winding element

\[{\tt e}_i = X^{i-1}Y^{k-2-(i-1)}\{0,\infty\} \in \bf M_k(N)\]

is of importance for the computation of special values. For any modular form \(f\in S_k(N)\) and homogeneous polynomial \(P(X,Y)\) of degree \(k-2\), let

\[\langle f, P(X,Y)\{0,\infty\}\rangle = -2\pi{}i \cdot \int_{0}^{i\infty} f(z) P(z,1){\rm dz}.\]

Fix a newform \(f \in S_k(N)\) corresponding to a space \(M\) of modular symbols, and let \(j\) be a integer in \(\{0,1,\ldots,k-1\}\). The winding element is significant because

\[L(f,j) = {(2\pi)^{j-1}\over i^{j+1}(j-1)!} \cdot \langle f, X^{j-1}Y^{k-2-(j-1)} \{0,\infty\}\rangle.\]

Moreover, the submodule that is generated by the winding element is used in the formula for a canonical rational part of the number \(L(M,j)\) (see LRatio, above).

WindingElement(M): ModSym -> ModSymElt

The winding element \(Y^{k-2}\{0,\infty\}\).

WindingElement(M, i): ModSym, RngIntElt -> ModSymElt

The winding element \(X^{i-1}Y^{k-2-(i-1)}\{0,\infty\}\).

TwistedWindingElement(M, i, eps): ModSym, RngIntElt, GrpDrchElt -> ModSymElt

The element \(\sum_{a \in ({\mathbb{Z}}/m{\mathbb{Z}})^*} \varepsilon(a)X^{i-1}Y^{k-2-(i-1)}\{0,{a\over{}m}\}\).

WindingLattice(M, j : parameters): ModSym, RngIntElt -> Lat

Bound: RngIntElt                    Default: -1

The image under RationalMapping(M) of the lattice generated by the images of the \(j\)th winding element under all Hecke operators \(T_n\). If \(M\) is the ambient space, then the image under RationalMapping(M) is not taken.

WindingSubmodule(M, j : parameters): ModSym, RngIntElt -> ModTupFld

Bound: RngIntElt                    Default: -1

The image under RationalMapping(M) of the vector space generated by all images of WindingElement(M,j) under all Hecke operators \(T_n\). If \(M\) is the ambient space, then the image under the rational period mapping is not taken.

TwistedWindingSubmodule(M, j, eps): ModSym, RngIntElt, GrpDrchElt -> ModTupFld

The Hecke submodule of the vector space \(\Phi(M)\) generated by the image of the \(j\)th \(\varepsilon\)-twisted modular winding element, where \(\Phi\) is RationalMapping(M). Some care is needed when using a modular symbol space in a \(+1\) or \(-1\) quotient.