Modular forms, modular symbols
(PARI-GP version 2.13.0)
Modular Forms
Dirichlet characters
Characters are encoded in three different ways:
•at INTD≡0,1 mod 4: the quadratic character (D/·);
• a t INTMOD Mod(m, q), m ∈ (Z/q)∗ using a canonical bijection with the dual group (the Conrey characterχq(m,·));
• a pair [G, chi], where G = znstar(q,1) encodes (Z/qZ)∗ = P
j≤k(Z/djZ)·gj and the vectorchi = [c1, . . . , ck] encodes the character such thatχ(gj) =e(cj/dj).
initializeG= (Z/qZ)∗ G = znstar(q,1)
convert datumDto [G, χ] znchar(D)
Galois orbits of Dirichlet characters chargalois(G) Spaces of modular forms
Arguments of the form [N, k, χ] give the level weight and nebenty- pusχ;χcan be omitted: [N, k] means trivialχ.
initializeSknew(Γ0(N), χ) mfinit([N, k, χ],0) initializeSk(Γ0(N), χ) mfinit([N, k, χ],1) initializeSkold(Γ0(N), χ) mfinit([N, k, χ],2) initializeEk(Γ0(N), χ) mfinit([N, k, χ],3) initializeMk(Γ0(N), χ) mfinit([N, k, χ])
find eigenforms mfsplit(M)
statistics on self-growing caches getcache() We letM =mfinit(. . .) denote a modular space.
describe the spaceM mfdescribe(M)
recover (N, k, χ) mfparams(M)
. . .the space identifier (0 to 4) mfspace(M) . . .the dimension ofM overC mfdim(M) . . .aC-basis (fi) ofM mfbasis(M) . . .a basis (Fj) of eigenforms mfeigenbasis(M) . . .polynomials definingQ(χ)(Fj)/Q(χ) mffields(M) matrix of Hecke operatorTnon (fi) mfheckemat(M, n) eigenvalues ofwQ mfatkineigenvalues(M, Q) basis of period poynomials for weightk mfperiodpolbasis(k) basis of the Kohnen +-space mfkohnenbasis(M) . . .new space and eigenforms mfkohneneigenbasis(M, b) isomorphismSk+(4N, χ)→S2k−1(N, χ2) mfkohnenbijection(M) Useful data can also be obtained a priori, without computing a complete modular space:
dimension ofSknew(Γ0(N), χ) mfdim([N, k, χ]) dimension ofSk(Γ0(N), χ) mfdim([N, k, χ],1) dimension ofSkold(Γ0(N), χ) mfdim([N, k, χ],2) dimension ofMk(Γ0(N), χ) mfdim([N, k, χ],3) dimension ofEk(Γ0(N), χ) mfdim([N, k, χ],4) Sturm’s bound forMk(Γ0(N), χ) mfsturm(N, k) Γ0(N)cosets
list of right Γ0(N) cosets mfcosets(N) identify coset a matrix belongs to mftocoset Cusps
a cusp is given by a rational number oroo.
lists of cusps of Γ0(N) mfcusps(N) number of cusps of Γ0(N) mfnumcusps(N) width of cuspcof Γ0(N) mfcuspwidth(N, c) is cuspcregular forMk(Γ0(N), χ)? mfcuspisregular([N, k, χ], c)
Create an individual modular form
Besidesmfbasis and mfeigenbasis, an individual modular form can be identified by a few coefficients.
modular form from coefficients mftobasis(mf,vec) There are also many predefined ones:
Eisenstein seriesEk onSl2(Z) mfEk(k) Eisenstein-Hurwitz series on Γ0(4) mfEH(k) unaryθfunction (for characterψ) mfTheta({ψ})
Ramanujan’s ∆ mfDelta()
Ek(χ) mfeisenstein(k, χ)
Ek(χ1, χ2) mfeisenstein(k, χ1, χ2) eta quotientQ
iη(ai,1·z)ai,2 mffrometaquo(a) newform attached to ell. curveE/Q mffromell(E) identify anL-function as a eigenform mffromlfun(L) θfunction attached toQ >0 mffromqf(Q) trace form inSknew(Γ0(N), χ) mftraceform([N, k, χ]) trace form inSk(Γ0(N), χ) mftraceform([N, k, χ],1) Operations on modular forms
In this section,f,gand theF[i] are modular forms
f×g mfmul(f, g)
f /g mfdiv(f, g)
fn mfpow(f, n)
f(q)/qv mfshift(f, v)
P
i≤kλiF[i],L= [λ1, . . . , λk] mflinear(F, L)
f=g? mfisequal(f,g)
expanding operatorBd(f) mfbd(f, d)
Hecke operatorTnf mfhecke(mf, f, n)
initialize Atkin–Lehner operatorwQ mfatkininit(mf, Q) . . .applywQtof mfatkin(wQ, f) twist by the quadratic char (D/·) mftwist(f, D)
derivative wrt.q·d/dq mfderiv(f)
seef over an absolute field mfreltoabs(f) Serre derivative
q·dqd −12kE2
f mfderivE2(f)
Rankin-Cohen bracket [f, g]n mfbracket(f, g, n) Shimura lift off for discriminantD mfshimura(mf, f, D) Properties of modular forms
In this section,f=P
nfnqn is a modular form in some spaceM with parametersN, k, χ.
describe the formf mfdescribe(f)
(N, k, χ) for formf mfparams(f)
the space identifier (0 to 4) forf mfspace(mf, f)
[f0, . . . , fn] mfcoefs(f, n)
fn mfcoef(f, n)
isfa CM form? mfisCM(f)
isfan eta quotient? mfisetaquo(f)
Galois rep. attached to all (1, χ) eigenforms mfgaloistype(M) . . .single eigenform mfgaloistype(M, F) . . .as a polynomial fixed by KerρF mfgaloisprojrep(M, F) decomposefonmfbasis(M) mftobasis(M, f) smallest level on whichfis defined mfconductor(M, f) decomposefon⊕Snewk (Γ0(d)),d|N mftonew(M, f) valuation offat cuspc mfcuspval(M, f, c) expansion at∞off|kγ mfslashexpansion(M, f, γ, n) n-Taylor expansion off ati mftaylor(f, n) all rational eigenforms matching criteria mfeigensearch . . .forms matching criteria mfsearch
Forms embedded into C
Given a modular formfinMk(Γ0(N), χ) its field of definitionQ(f) hasn= [Q(f) :Q(χ)] embeddings into the complex numbers. If n= 1, the following functions return a single answer, attached to the canonical embedding off inC[[q]]; else a vector of nresults, corresponding to thenconjugates off.
complex embeddings ofQ(f) mfembed(f)
... embed coefs off mfembed(f, v)
evaluatef atτ∈ H mfeval(f, τ)
L-function attached tof lfunmf(mf, f) . . .eigenforms of new spaceM lfunmf(M) Periods and symbols
The functions in this section depend on [Q(f) : Q(χ)] as above.
initialize symbolf sattached tof mfsymbol(M, f) evaluate symbolf son pathp mfsymboleval(f s, p) Petersson product offandg mfpetersson(f s, gs) period polynomial of formf mfperiodpol(M, f s) period polynomials for eigensymbolF S mfmanin(F S)
Modular Symbols
LetG = Γ0(N), Vk = Q[X, Y]k−2, Lk =Z[X, Y]k−2. We let
∆ = Div0(P1(Q)); an element of ∆ is a path between cusps of X0(N) via the identification [b]−[a]→the path fromatob. A path is coded by the pair [a, b], wherea, bare rationals oroo, de- noting the point at infinity (1 : 0).
Let Mk(G) = HomG(∆, Vk) ' Hc1(X0(N), Vk); an element of Mk(G) is aVk-valuedmodular symbol. There is a natural decom- positionMk(G) =Mk(G)+⊕Mk(G)−under the action of the∗ involution, induced by complex conjugation. Themsinitfunction computes eitherMk (ε= 0) or its±-parts (ε=±1) and fixes a minimal set ofZ[G]-generators (gi) of ∆.
initializeM=Mk(Γ0(N))ε msinit(N, k,{ε= 0})
the levelM msgetlevel(M)
the weightk msgetweight(M)
the signε msgetsign(M)
Farey symbol attached toG mspolygon(M) . . .attached toH < G msfarey(F, inH)
H\Gand rightG-action mscosets(genG, inH)
Z[G]-generators (gi) and relations for ∆ mspathgens(M) decomposep= [a, b] on the (gi) mspathlog(M, p) Create a symbol
Eisenstein symbol attached to cuspc msfromcusp(M, c) cuspidal symbol attached toE/Q msfromell(E) symbol having given Hecke eigenvalues msfromhecke(M, v,{H})
issa symbol ? msissymbol(M, s)
Operations on symbols
the list of alls(gi) mseval(M, s)
evaluate symbolson pathp= [a, b] mseval(M, s, p) Petersson product ofsandt mspetersson(M, s, t) Operators on subspaces
An operator is given by a matrix of a fixedQ-basis.H, if given, is a stableQ-subspace ofMk(G): operator is restricted toH. matrix of Hecke operatorTporUp mshecke(M, p,{H}) matrix of Atkin-LehnerwQ msatkinlehner(M, Q{H}) matrix of the∗involution msstar(M,{H}) Subspaces
A subspace is given by a structure allowing quick projection and restriction of linear operators. Its fist component is a matrix with integer coefficients whose columns for aQ-basis. IfHis a Hecke- stable subspace ofMk(G)+orMk(G)−, it can be split into a direct sum of Hecke-simple subspaces. To a simple subspace corresponds a single normalized newformP
nanqn.
cuspidal subspaceSk(G)ε mscuspidal(M) Eisenstein subspaceEk(G)ε mseisenstein(M)
new part ofSk(G)ε msnew(M)
splitH into simple subspaces (of dim≤d) mssplit(M, H,{d})
dimension of a subspace msdim(M)
(a1, . . . , aB) for attached newform msqexpansion(M, H,{B}) Z-structure fromH1(G, Lk) on subspaceA mslattice(M, A)
Overconvergent symbols andp-adicLfunctions
LetM be a full modular symbol space given by msinitandpbe a prime. To a classical modular symbolφof levelN (vp(N)≤1), which is an eigenvector forTpwith nonzero eigenvalueap, we can attach ap-adicL-functionLp. The functionLpis defined on con- tinuous characters of Gal(Q(µp∞)/Q); in GP we allow characters hχis1τs2, where (s1, s2) are integers,τis the Teichm¨uller character andχis the cyclotomic character.
The symbolφcan be lifted to anoverconvergentsymbol Φ, taking values in spaces ofp-adic distributions (represented in GP by a list of moments modulopn).
mspadicinitprecomputes data used to lift symbols. Ifflagis given, it speeds up the computation by assuming that vp(ap) = 0 if flag = 0 (fastest), and that vp(ap) ≥ flag otherwise (faster as flagincreases).
mspadicmomentscomputes distributionsmuattached to Φ allowing to computeLp to high accuracy.
initializeMpto lift symbols mspadicinit(M, p, n,{flag})
lift symbolφ mstooms(Mp, φ)
eval overconvergent symbol Φ on pathp msomseval(Mp,Φ, p) muforp-adicL-functions mspadicmoments(Mp, S,{D= 1}) L(r)p (χs),s= [s1, s2] mspadicL(mu,{s= 0},{r= 0}) Lˆp(τi)(x) mspadicseries(mu,{i= 0})
Based on an earlier version by Joseph H. Silverman October 2020 v2.37. Copyright c2020 K. Belabas
Permission is granted to make and distribute copies of this card provided the copyright and this permission notice are preserved on all copies.
Send comments and corrections tohKarim.Belabas@math.u-bordeaux.fri
