# 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 [[Stein, 2000](../../references.md#cite-stein-phd)]. ## `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` ```magma 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 (ex-75fe6d)` ```magma > 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 ``` ## 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` ```magma 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` ```magma 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.