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The LSeries command creates the L-series L(A, s)
associated to a modular abelian variety A over Q
or a cyclotomic field. No actual computation is performed.
The L-series associated to the abelian variety A.
> 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.
> 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)
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).
The abelian variety the L-series L is associated to.
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).
> 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
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).
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.
Factored: BoolElt Default: false
The characteristic polynomial of (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.
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.
> 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);
[
<x^4 - 2*x^3 + 14*x^2 - 16*x + 64, 1>,
<x^4 + x^3 + 3*x^2 + 2*x + 4, 1>,
<x^4 + 3*x^3 + 5*x^2 + 6*x + 4, 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.
> A := ChangeRing(JZero(23),GF(2^10));
> NumberOfRationalPoints(A);
1073875 1073875
> Factorization($1);
[ <5, 3>, <11, 2>, <71, 1> ]
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.
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(L, s) : ModAbVarLSer, RngIntElt -> FldRatElt
Given an abelian variety A over 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)!/((2π )s - 1 * Ω s), where s is a
"critical integer",
and Ω 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.
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.
First we demonstrate each evaluation command for the L-series
of J 0(23).
> 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 Δ.
> 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).
> 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.
> LRatio(J(1),1);
1/11
> LRatio(J(2),1);
1/1382426761
Each simple factor has nonzero LRatio, so L(J, 1) != 0.
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.
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.
> 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.
> LeadingCoefficient(LSeries(J(1)),1,100);
1.419209649338215616003188084281 1
> LeadingCoefficient(LSeries(J(5)),1,100);
1.228051952859142052034769858445 3
We give a few more examples.
> 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
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