Computing coefficients of modular forms
(Work in progress; extension of results of Couveignes, Edixhoven et al.)
Peter Bruin
Mathematisch Instituut, Universiteit Leiden
Th ´eorie des nombres et applications CIRM, Luminy, 30 November 2009
Introduction
Let
k
andn
be positive integers, and letf
be a modular form of weightk
forΓ
1(n)
, withq
-expansionf =
X
m≥0
a
m(f )q
m.
It is known that
f
is determined by thea
m(f )
form ≤
k
12
[SL
2(Z) : {±1}Γ
1(n)].
Question: given these
a
m(f )
, is it possible to efficiently computea
m(f )
for largem
?This only seems reasonable to ask when given the factorisation of
m
: the recurrence relations for thea
m(f )
suggest that one could otherwise factor products of two large prime factors efficiently.Introduction
We may assume
f
is a Hecke eigenform, normalised such thata
1(f ) = 1
.Theorem 1 (tentative for
n > 1
): There is a (probabilistic) algorithm that, given positive integersk
andn
withn
square-free, a normalised eigenformf
of weightk
forΓ
1(n)
, and an integerm > 0
in factored form, computesa
m(f )
. If the generalised Riemann hypothesis for number fields is true, the algorithm runs in time polynomial ink
,n
andlog m
.For
n = 1
: proved by J.-M. Couveignes, S. J. Edixhoven, R. de Jong and F. Merkl (preprint, 2006/2009; to appear in the Ann. Math. Studies series).For
n > 1
: work in progress, to appear in my thesis (2010).Note: Our algorithm runs in time polynomial time in the input size, whereas
existing algorithms (modular symbols) are exponential in
log m
.Reduction to eigenforms over finite fields
By the recurrence relation expressing
a
m(f )
in thea
p(f )
forp | m
prime, we are reduced to the problem of computinga
p(f )
for prime numbersp
.Write
Q(f )
for the number field generated by thea
m(f )
. By Deligne’s bound|σ(a
p(f ))| ≤ 2p
(k−1)/2 for allσ: Q(f ) → C
, it suffices to computea
p(f )
modulo sufficiently many small primesλ
ofQ(f )
.Remark: We need the generalised Riemann hypothesis to ensure the
exis-tence of a sufficient supply of such
λ
, uniformly inQ(f )
.Theorem 2 (tentative for
n > 1
): There exists a (probabilistic) algorithm that, given positive integersk
andn
withn
square-free, a normalised eigenformf
over a finite fieldF
and a prime numberp
, computesa
p(f )
in time polyno-mial ink
,n
andlog p
.Modular Galois representations
The strategy for computing
a
p(f )
for an eigenformf
over a finite fieldF
is to compute the Galois representation associated tof
.Let
l
be the characteristic ofF
. There exists a unique semi-simple continuous representationρ
f: Gal( Q /Q) → GL
2(F)
that is unramified outside
nl
and such that the Frobenius conjugacy class at a primep - nl
has characteristic polynomialt
2− a
p(f )t + (p)p
k−1∈ F[t].
In particular,
a
p(f )
is the trace of a Frobenius atp
underρ
f .What we want to compute
Let
E
f be the finite Galois extension ofQ
such thatρ
f factors asρ
f: Gal( Q /Q) Gal(E
f/Q) GL
2(F).
Then by computing
ρ
f we mean producing the following data:•
the multiplication table ofE
f with respect to someQ
-basis(b
1, . . . , b
r)
ofE
f ;•
for every elementσ ∈ Gal(E
f/Q)
, the matrix ofσ
with respect to the basis(b
1, . . . , b
r)
and the elementρ
f(σ) ∈ GL
2(F)
.If
ρ
f is reducible, then it is associated to an Eisenstein series and is easy to compute.Modular Galois representations in Jacobians
From now on we assume that
ρ
f: Gal( Q /Q) → GL
2(F)
is irreducible. Af-ter twistingρ
f by a power of the cyclotomic character, we may assume more-over that2 ≤ k ≤ l + 1.
Finally we may assume that
F
is generated by thea
m(f )
.Notation:
n
0=
n
ifk = 2
;nl
ifk > 2
;X
1(n
0) =
modular curve forΓ
1(n
0)
-structures;J
1(n
0) =
Jacobian ofX
1(n
0);
g = genus(X
1(n
0)) = dim(J
1(n
0)).
Modular Galois representations in Jacobians
Let
T
1(n
0) ⊆ End J
1(n
0)
denote the Hecke algebra. By the work of various people (Mazur, Ribet, Gross,. . .
) there is a surjective homomorphismT
1(n
0) → F
T
m7→ a
m(f ).
Let
m
⊂ T
1(n
0)
be its kernel. Then theF[Gal( Q /Q)]
-moduleJ
1(n
0)[m]( Q )
is non-zero and ‘usually’ isomorphic to
ρ
f (in general it has a composition chain consisting of copies ofρ
f ).Strategy for computing Galois representations
To find
ρ
f , we are going to explicitly compute theF
-vector space schemeJ
1(n
0)[m]
overQ
. We do this by choosing a suitable closed immersionι: J
1(n
0)[m] A
1Q.
The image of
ι
is defined by some non-zero polynomialP
ι∈ Q[x]
.By “explicitly computing
J
1(n
0)[m]
” we mean computingP
ι together with a collection of ring homomorphisms defining theF
-vector space scheme struc-ture onSpec Q[x]/(P
ι)
.From these data we can compute
ρ
f by standard methods (mostly factorisa-tion of polynomials overQ
).Choosing a suitable map
Fix a point
O ∈ X
1(n
0)(Q)
. Assume for simplicity that the Abel–Jacobi mapSym
gX
1(n
0) J
1(n
0)
D 7→ [D − gO]
is an isomorphism above
J
1(n
0)[m]
. We choose a rational functionψ: X
1(n
0) → P
1(Q)
(e.g. a quotient of two modular forms of the same weight) Then we obtain a map
ψ
∗: Sym
gX
1(n
0) → Sym
gP
1Q∼
= P
gQ.
We choose
ψ
such thatψ
∗ is a closed immersion on the inverse image ofJ
1(n
0)[m]
under the Abel–Jacobi map.Choosing a suitable map
We next choose a suitable rational map
λ: P
gQ99K A
1Q⊂ P
1Qthat is a quotient of linear forms. We define our closed immersion
ι: J
1(n
0)[m] A
1Qas the arrow making the diagram
Sym
gX
1(n
0)
J
1(n
0)
⊃
J
1(n
0)[m]
ψ∗
y
Sym
gP
1Q−→
∼P
gQ99K
λA
1Qcommutative.
How to compute
P
ιRecall that we want to compute (among other things) the polynomial
P
ι defin-ing the image of the closed immersionι: J
1(n
0)[m] A
1Q.
To compute
P
ι , we use numerical approximation together with a bound on the heights of the coefficients ofP
ι ,The polynomial
P
ι can be approximated either using computations over the complex numbers (deterministically) or modulo many small prime numbers (probabilistically).How to compute
P
ι modulo prime numbersFor computing
P
ι modulo a prime numberp
, one needs to be able to compute in the Jacobian ofX
1(n
0)
over finite fields of characteristicp
: picking random elements, computing the Frobenius map, evaluating Hecke operators, etc.For
n = 1
, one can use a (singular) plane model ofX
1(5l)
overF
p with singularities and apply algorithms by Couveignes for computing in the Jacobian of such a curve.For
n ≥ 1
, one can use the projective embedding ofX
1(n
0)
defined by modular forms of weight 2 and use algorithms of K. Khuri-Makdisi, Couveignes (adapted to this situation), C. Diem and myself.Computing in Jacobians of projective curves over finite fields
For
n ≥ 5
, the line bundleL = ω
2 of modular forms of weight 2 onX
1(n)
over a fieldK
(of characteristic not dividingn
) gives a closed immersionX
1(n)
KPΓ(X
1(n), L).
An effective divisor
D
withdeg D = deg L
can be represented as the sub-spaceΓ(X
1(n), L
2(−D))
. To such aD
we associate the point[L(−D)]
ofJ
1(n)(K)
.Khuri-Makdisi has developed algorithms for computing with elements of the Jacobian represented in this way. Based in part on work of Couveignes and of Diem, I have shown that if
K
is finite, one can also compute Frobenius maps, Hecke correspondences, Kummer maps and Frey–R ¨uck pairings. These can be used to computeJ
1(n
0)[m]
(i.e. computeP
ι ) modulo prime numbers.Height bounds
We use Arakelov intersection theory on the arithmetic surface
X
1(n
0)
to find bounds for the heights of the coefficients of the polynomialP
ι . Intersection numbers at infinite places can be expressed in terms of canonical Green func-tions of the Riemann surfacesX
1(n
0)(C)
.We need to study the semi-stable reduction of
X
1(n
0)
, and find bounds for canonical Green functions and for sup-norms of modular forms. Work of J. Jor-genson and J. Kramer, using spectral theory of automorphic forms for Fuchsian groups, implies that the latter quantities are bounded independently ofn
0 .Using methods similar to that of Jorgenson and Kramer, I have found bounds that could fairly easily be made explicit. These methods can be interpreted as based on the fact that the Green function is the constant term of the resolvent kernel of the Laplace operator.
Application: explicit realisations of Galois groups
The complex analytic method for computing modular Galois representations has been used by J. Bosman to compute various explicit polynomials over
Q
whose splitting fields have interesting Galois groups, such asSL
2(F
16)
andPSL
2(F
49)
.The algorithm is so far only practical in small cases. Instead of using explicit height bounds, Bosman verified the results using the fact that Serre’s conjec-ture is true.
We expect that combining these complex analytic algorithms with the algo-rithms over finite fields (to be implemented) can be used to compute explicit realisations of more Galois groups.
Application: representation numbers of lattices
Let
L
be an even integral lattice of rankk
, writer
L(m) = #{x ∈ L | (x, x) = m},
and let
θ
L=
X
m≥0
r
L(2m)q
m∈ Z[q]
be the
θ
-series ofL
. Thenθ
L is theq
-expansion of a modular form of weightk/2
forΓ
1(n)
, wheren
is the level ofL
.Example: the Leech lattice (rank 24 and level 1). We know how to write its
θ
-series as a linear combination ofE
12 and∆
. Its representation numbers can therefore be computed by the work of Couveignes, Edixhoven, de Jong and Merkl.Sums of squares
For
L = Z
k with(x, y) = 2
P
ki=1x
iy
i , theθ
-seriesθ
Zk is a modularform of weight
k/2
for the groupΓ
1(4)
. From the identityθ
Zk= (θ
Z)
k= (1 + 2q + 2q
4+ 2q
9+ · · ·)
kwe can quickly compute the first few coefficients of
θ
Zk . This has the followingapplication to representing integers as sums of squares.
Expected theorem 3: There exists a (probabilistic) algorithm that, given an
even integer
k ≥ 0
and an integerm ≥ 1
in factored form, computes the number of ways in whichm
can be written as a sum ofk
squares, and that runs in time polynomial ink
andlog m
under the generalised Riemann hy-pothesis for number fields.A question about lattices
To compute coefficients of
θ
-series of a lattice in this way, one needs to input the first few coefficients of theθ
-series into the algorithm.It is known that finding the length of the shortest vector in a lattice is already a hard problem, but the proof of this does not seem to involve the level of the lattice. This raises the following question about counting short vectors in lattices.
Question: Does there exist an algorithm that, given a lattice