# $L$-series ## Creation The `LSeries` command creates the $L$-series $L(A,s)$ associated to a modular abelian variety $A$ over ${\mathbb{Q}}$ or a cyclotomic field. No actual computation is performed. ### `LSeries(A): ModAbVar -> ModAbVarLSer` The $L$-series associated to the abelian variety $A$. ### `Example: Lser Creation (ex-1dae21)` ```magma > A := JZero(23); > L := LSeries(A); > L; L(JZero(23),s): L-series of Modular abelian variety JZero(23) of dimension 2 and level 23 over Q > LSeries(ModularAbelianVariety("65B")); L(65B,s): L-series of Modular abelian variety 65B of dimension 2 and level 5*13 over Q ``` You can create $L$-series of abelian varieties over cyclotomic fields, but currently no interesting functionality is implemented for them. ```magma > LSeries(BaseExtend(JZero(11),CyclotomicField(5))); L(JZero(11),s): L-series of Modular abelian variety JZero(11) of dimension 1 and level 11 over Q(zeta_5) ``` ## Invariants ### `CriticalStrip(L): ModAbVarLSer -> RngIntElt, RngIntElt` Let $L$ be the $L$-function of some modular abelian variety $A$. This function returns integers $x$ and $y$ such that the critical strip for $L$ is the set of complex numbers with real part strictly between $x$ and $y$. If $W$ is the set of weights of newforms that give rise to factors of $A$, then this command returns $0$ and `Max(W)`. ### `ModularAbelianVariety(L): ModAbVarLSer -> ModAbVar` The abelian variety the $L$-series $L$ is associated to. ### `Example: Lser Invariants (ex-c8a997)` We define several $L$-functions of modular abelian varieties and modular motives, and compute their critical strip (which is from $0$ to $k$, where $k$ is the weight). ```magma > L := LSeries(JZero(37)); > CriticalStrip(L); 0 2 > L := LSeries(JZero(37,6)); > CriticalStrip(L); 0 6 > J := JOne(11,3); J; Modular motive JOne(11,3) of dimension 5 and level 11 over Q > CriticalStrip(LSeries(J)); 0 3 > A_delta := JZero(1,12); > L := LSeries(A_delta); > CriticalStrip(L); 0 12 > ModularAbelianVariety(L); Modular motive JZero(1,12) of dimension 1 and level 1 over Q ``` ## Characteristic Polynomials of Frobenius Elements Let $A$ be a modular abelian variety. The characteristic polynomials of Frobenius elements acting on the $\ell$-adic Tate modules of $A$ define the local $L$-factors of $L(A,s)$. ### `FrobeniusPolynomial(A : parameters): ModAbVar -> RngUPolElt` ```magma Factored: BoolElt Default: false ``` The characteristic polynomial of Frobenius on the abelian variety $A$ defined over a finite field. If `Factored` is set to `true`, return a factorization instead of a polynomial. ### `FrobeniusPolynomial(A, p : parameters): ModAbVar, RngIntElt -> RngUPolElt` ```magma Factored: BoolElt Default: false ``` The characteristic polynomial of ${\rm Frob}_p$ acting on any $\ell$-adic Tate module of the abelian variety $A$ over a number field, where $p$ and $\ell$ do not divide the level of $A$. If the base ring has degree bigger than $1$, then return a sequence of characteristic polynomials, one for each prime lying over $p$, sorted by degree. If `Factored` is set to `true`, return a factorization instead of a polynomial. ### `FrobeniusPolynomial(A, P): ModAbVar, RngOrdIdl -> RngUPolElt` The characteristic polynomial of Frobenius at the nonzero prime ideal $P$ of a number field on the modular abelian variety $A$, where $P$ is assumed to be a prime of good reduction for $A$, and $A$ is defined over a field that contains the prime $P$. ### `Example: Lser Characteristic Polynomials Of Frobenius Elements (ex-671a3a)` ```magma > A := JZero(23); > FrobeniusPolynomial(A,2); x^4 + x^3 + 3*x^2 + 2*x + 4 > A := JZero(23) * JZero(11,4) * JOne(13); > FrobeniusPolynomial(A,2); x^12 + 2*x^11 + 17*x^10 + 40*x^9 + 145*x^8 + 362*x^7 + 798*x^6 + 1408*x^5 + 2104*x^4 + 2528*x^3 + 2528*x^2 + 1792*x + 1024 > Factorization($1); [ , , ] > A := BaseExtend(JZero(23),CyclotomicField(22)); > FrobeniusPolynomial(A,2); [ x^4 + 25*x^3 - 327*x^2 + 25600*x + 1048576 ] ``` These characteristic polynomials are used in the algorithm to compute the number of points on modular abelian varieties over finite fields. ```magma > A := ChangeRing(JZero(23),GF(2^10)); > NumberOfRationalPoints(A); 1073875 1073875 > Factorization($1); [ <5, 3>, <11, 2>, <71, 1> ] ``` ## Values at Integers in the Critical Strip Magma allows evaluation of $L$-series at integers lying within the critical strip. There exist algorithms for computing $L(A,s)$ for any complex number $s$, but these are not currently implemented in Magma. ### `L(s): RngIntElt, ModAbVarLSer -> RngElt` ### `Evaluate(L, s): ModAbVarLSer, RngIntElt -> FldReElt` ### `Evaluate(L, s, prec): ModAbVarLSer, RngIntElt, RngIntElt -> FldReElt` The value of $L$-series $L$ at $s$, where $s$ must be an integer that lies in the critical strip for $L$, computed using $prec$ terms of the power series or $100$ if $prec$ is not given. The power series used are the $q$-expansions of modular forms corresponding to differentials on $A$. It is not clear, a priori, what the relation is between $prec$ and the precision of the real number output by this command. (It is theoretically possible to give bounds, but we have not done this.) In practice, one can increase $prec$ and see how the output result changes. ### `LRatio(A, s): ModAbVar, RngIntElt -> FldRatElt` ### `LRatio(L, s): ModAbVarLSer, RngIntElt -> FldRatElt` Given an abelian variety $A$ over ${\mathbb{Q}}$ attached to a newform or an $L$-series $L$ of an abelian variety and an integer $s$, return the ratio $L(A,s)*(s-1)!/((2\pi )^{s-1}*\Omega _s)$, where $s$ is a “critical integer”, and $\Omega _s$ is the integral (Neron) volume of the group of real points on the optimal quotient $A'$ associated to $A$ when $s$ is odd, and the volume of the $-1$ eigenspace for conjugation when $s$ is even. ### `IsZeroAt(L, s): ModAbVarLSer, RngIntElt -> BoolElt` Given an $L$-series $L$ of a modular abelian variety and an integer $s$ in the critical strip for $L$ return `true` is $L(A, s)$ is zero. In contrast to the output of the `Evaluate` command above, the result returned by this command is provably correct. ### `Example: Lser Values At Integers In The Critical Strip (ex-88991d)` First we demonstrate each evaluation command for the $L$-series of $J_0(23)$. ```magma > L := LSeries(JZero(23)); > L(1); 0.248431866590599284683305769290 + 0.E-29*i > Evaluate(L,1); 0.248431866590599284683305769290 + 0.E-29*i > Evaluate(L,1,200); 0.248431866590599681207250339074 + 0.E-29*i > LRatio(L,1); 1/11 > L := LSeries(JZero(23)); > L(1); 0.248431866590599284683305770476 + 0.E-29*i > Evaluate(L,1,200); 0.248431866590599681207250340144 + 0.E-29*i ``` Next we compute the $L$-series of the motive attached to the weight $12$ level $1$ modular form $\Delta$. ```magma > A := JZero(1,12); > L := LSeries(A); > Evaluate(L,1); 0.0374412812685155417387703158443 > L(5); 0.66670918843400364382613022164 > Evaluate(L,1,200); 0.0374412812685155417387703158443 > LRatio(L,1); 11340/691 > LRatio(L,2); 24 > LRatio(L,3); 7 ``` We compute some ratios for $J_1(N)$ and factors of $J_1(N)$. ```magma > LRatio(JOne(13),1); 1/361 > J := JOne(23); > Evaluate(LSeries(J),1); 0.000000080777697074785775420090700066 + 0.000000053679621277482217773207669332*i ``` It looks kind of like $L(J_1(23),1)$ is zero. However, this is not the case! We can not compute `LRatio` for $J_1(23)$, since it not attached to a newform. We can, however, compute `LRatio` for each simple factor. ```magma > LRatio(J(1),1); 1/11 > LRatio(J(2),1); 1/1382426761 ``` Each simple factor has nonzero `LRatio`, so $L(J,1)\neq 0$. ## Leading Coefficient The $L$-function $L(A,s)$ has a Taylor expansion about any critical integer. The leading coefficient and order of vanishing of $L(A,s)$ about a critical integer can be computed. ### `LeadingCoefficient(L, s, prec): ModAbVarLSer, RngIntElt, RngIntElt -> FldReElt, RngIntElt` Given an $L$-series $L$ associated to a modular abelian variety $A$ and an integer $s$ in the critical strip for $L$ return the leading coefficient of the Taylor expansion about $s$ and the order of vanishing of $L$ at $s$. At present, $A$ must have weight $2$ and trivial character (so $s=1)$. It does not have to be attached to a newform. The argument $prec$ is the number of terms of the power series which are used. ### `Example: Lser Leading Coefficient (ex-54429e)` ```magma > LeadingCoefficient(LSeries(JZero(37)),1,100); 0.244264064925838981349867782965 1 > LeadingCoefficient(LSeries(JZero(37)(1)) ,1,100); 0.305999773800085290044094075725 1 > J := JZero(3^5); > LeadingCoefficient(LSeries(J),1,100); 15.140660788463628991688955015326 + 0.E-27*i 4 ``` The order of vanishing of $4$ for $J_0(3^5)$ comes from an elliptic curve and a $3$-dimensional abelian variety that have order of vanishing $1$ and $3$, respectively. ```magma > LeadingCoefficient(LSeries(J(1)),1,100); 1.419209649338215616003188084281 1 > LeadingCoefficient(LSeries(J(5)),1,100); 1.228051952859142052034769858445 3 ``` ### `Example: Lser Leading Coefficient2 (ex-b1fb66)` We give a few more examples. ```magma > L := LSeries(ModularAbelianVariety("389A",+1)); > LeadingCoefficient(L,1,100); 0.75931650029224679065762600319 2 > > A := JZero(65)(2); A; Modular abelian variety 65B of dimension 2, level 5*13 and conductor 5^2*13^2 over Q > L := LSeries(A); > LeadingCoefficient(L,1,100); 0.91225158869818984109351402175 + 0.E-29*i 0 > A := JZero(65)(3); A; Modular abelian variety 65C of dimension 2, level 5*13 and conductor 5^2*13^2 over Q > L := LSeries(A); > LeadingCoefficient(L,1,100); 0.452067921768031069917486135000 + 0.E-29*i 0 ```