Modular Hecke Algebras and their Hopf Symmetry - Alain Connes

60 Pages·2012·0.32 MB·English

Modular Hecke Algebras and their
Hopf Symmetry
Alain Connes Henri Moscovici(cid:3)
Coll(cid:18)ege de France Department of Mathematics
3 rue d’Ulm The Ohio State University
75005 Paris, France Columbus, OH 43210, USA
Dedicated to Pierre Cartier
Abstract
Weintroduceandbegintoanalyseaclassofalgebras,associatedto
congruence subgroups,that extendboththe algebra of modularforms
of all levels and the ring of classical Hecke operators. At the intuitive
level, these arealgebras of‘polynomial coordinates’ forthe‘transverse
space’ of lattices modulo the action of the Hecke correspondences.
Their underlying symmetry is shown to be encoded by the same Hopf
algebra that controls the transverse geometry of codimension 1 folia-
tions. Its action is shown to span the ‘holomorphic tangent space’ of
the noncommutative space, and each of its three basic Hopf cyclic co-
cycles acquires a speci(cid:12)c meaning. The Schwarzian 1-cocycle gives an
innerderivation implemented by thelevel 1 Eisenstein series ofweight
4. The Hopf cyclic 2-cocycle representing the transverse fundamental
class provides a natural extension of the (cid:12)rst Rankin-Cohen bracket
to the modular Hecke algebras. Lastly, the Hopf cyclic version of the
Godbillon-Vey cocycle gives rise to a 1-cocycle on PSL(2;Q) with val-
ues in Eisenstein series of weight 2, which, when coupled with the
‘period’ cocycle, yields a representative of the Euler class.
(cid:3)
Research supported by the National Science Foundation award no. DMS-9988487.
1
Introduction
The aim of this paper is to introduce and begin to analyse a class of al-
gebras, called modular Hecke algebras, which encode the two a priori unre-
lated structures on modular forms, namely the algebrastructure given by the
pointwise product on the one hand, and the action of the Hecke operators on
the other. We associate to any congruence subgroup (cid:0) of SL(2;Z) a crossed
product algebra A((cid:0)), the modular Hecke algebra of level (cid:0), which is a direct
extension of both the ring of classical Hecke operators and of the algebra
M((cid:0)) of (cid:0)-modular forms. With M denoting the algebra of modular forms
of arbitrary level, the elements of A((cid:0)) are maps with (cid:12)nite support
F : (cid:0)nGL+(2;Q) ! M; (cid:0)(cid:11) 7! F 2 M;
(cid:11)
satisfying the covariance condition
F = F j(cid:13); 8(cid:11) 2 GL+(2;Q);(cid:13) 2 (cid:0)
(cid:11)(cid:13) (cid:11)
and their product is given by convolution. In the simplest case (cid:0)(1) =
SL(2;Z), the elements of A((cid:0)(1)) are encoded by a (cid:12)nite number of modular
forms f 2 M((cid:0) (N)) of arbitrary high level and the product operation is
N 0
non-trivial. The algebra A((cid:0)) acts on M((cid:0)) extending the action of classical
Hecke operators, and the ‘cuspidal’ elements form a two-sided ideal of A((cid:0)).
Our starting point is the basic observation that the Hopf algebra H , which
1
was discovered (cf. [12]) in the analysis of codimension 1 foliations as under-
lying symmetry of the ‘transverse’ geometry, admits a natural action on the
crossed product algebras A((cid:0)). This action is su(cid:14)ciently non-trivial to span
the ‘tangent space’ of the non-commutative space Q((cid:0)) with coordinate ring
A((cid:0)), and is thus a key ingredient in the understanding of the geometry of
Q((cid:0)). At an intuitive level these non-commutative spaces are obtained from
the quotient of the space of lattices by the Hecke correspondences, while
A((cid:0)) is the simplest algebra of polynomial coordinates on Q((cid:0)).
To describe the action of H , we recall that, as an algebra, H coincides with
1 1
the universal enveloping algebra of the Lie algebra with basis fX;Y;(cid:14) ;n (cid:21)
n
1g and brackets
[Y;X] = X; [Y;(cid:14) ] = n(cid:14) ; [X;(cid:14) ] = (cid:14) ; [(cid:14) ;(cid:14) ] = 0; n;k;‘ (cid:21) 1;
n n n n+1 k ‘
2
whilethecoproductwhich confers ittheHopfalgebrastructure isdetermined
by the identities
(cid:1)Y = Y (cid:10)1+1(cid:10)Y ; (cid:1)(cid:14) = (cid:14) (cid:10)1+1(cid:10)(cid:14) ;
1 1 1
(cid:1)X = X (cid:10)1+1(cid:10)X +(cid:14) (cid:10)Y ;
1
together with the property that (cid:1) : H ! H (cid:10) H is an algebra homo-
1 1 1
morphism. The action of X on A((cid:0)) is given by a classical operator going
back to Ramanujan ([25], [27]), which corrects the usual di(cid:11)erentiation by
the logarithmic derivative of the Dedekind eta function (cid:17)(z). The action of
Y is given by the standard grading by the weight (the Euler operator) on
modular forms. Finally, (cid:14) and its higher ‘derivatives’ (cid:14) act by generalized
1 n
cocycles on GL+(2;Q) with values in modular forms.
The main result of the present paper is a rather complete understanding of
the above action of H , viewed as the underlying symmetry of the noncom-
1
mutative space Q((cid:0)) obtained from the quotient of the space of lattices by
theHecke correspondences. The picturethatemerges isthatofasurprisingly
close analogy between the action of Hecke operators on modular forms and
the action of a discrete subgroup of Di(cid:11)(S1) on polynomial functions on the
frame bundle of S1. The role of the angular variable (cid:18) 2 R=2(cid:25)Z is assumed
by the simplest Eichler integral, namely the primitive Z 2 C=(cid:3) of the holo-
morphic di(cid:11)erential form 2(cid:25)i (cid:17)4dz on the elliptic curve X (1) = (cid:0)(1)nH ,
6 0 0 (cid:3)
where (cid:0)(1) is the commutator subgroup of (cid:0)(1). The lattice 2(cid:25)Z (cid:26) R is
0
replaced by the equilateral lattice (cid:3) (cid:26) C of periods of Z. The part of the
circle
ei(cid:18) = cos(cid:18) + i sin(cid:18); x2 + y2 = 1
isthusplayedbythegenus1curveX (1) (cid:24) C=(cid:3)withWeierstrassparametriza-
0
tion and equation
(cid:0)2E
(} (Z); } (Z)) = 3 j; 6 ; y2 = 4(x3 (cid:0) 1728)
(cid:3) 0(cid:3) (cid:17)12
(cid:18) (cid:19)
p
usingtraditionalnotations. Thedi(cid:11)eomorphisms (cid:30)((cid:18)) arereplacedbyHecke
transformations Zj0(cid:13), while the Jacobian (cid:30)0((cid:18)) becomes J((cid:13)) = dZdjZ0(cid:13).
Accordingly, the higher cocycles d n log(cid:30)((cid:18)) which allow us to de(cid:12)ne the
d(cid:18) 0
action of (cid:14) take the form d n logJ((cid:13)). Furthermore, the usual projective
n dZ (cid:0) (cid:1)
structure on S1, corresponding to t = tan((cid:18)), which in terms of (cid:18) is given
(cid:0) (cid:1) 2
3
by the quadratic di(cid:11)erential
1
(cid:26) := ft;(cid:18)gd(cid:18)2 = d(cid:18)2;
2
has as its modular counterpart the rational projective structure on X (1)
0
given by the quadratic di(cid:11)erential
xdx2 xdx2
$ = = :
0
2y2 8(x3 (cid:0)1728)
This Schwarzian relation between Z and $ is similar to the classical relation
0
between the modular invariant j and E in the case of genus zero [22].
4
With the above dictionary in mind, one can proceed to transfer transverse
geometry concepts and constructions to the setting of modular forms (cf.
Theorem 12 below). The basic tool is the cyclic cohomology of Hopf algebras
developped in ([12]). For H one gets three basic cyclic classes, (cid:14) , (cid:14) and
1 20 1
F which in the original action of H on the crossed product of functions
1
on the frame bundle on S1 by a discrete subgroup of Di(cid:11)(S1) correspond
respectively to
Schwarzian (cid:14) := (cid:14) (cid:0) 1(cid:14)2
20 2 2 1
Godbillon-Vey Class (cid:14)
1
Transverse Fundamental Class F := X (cid:10)Y (cid:0)Y (cid:10)X (cid:0)(cid:14) Y (cid:10)Y .
1
In the present paper we shall compute the above three classes in the action
of H on the modular Hecke algebras.
1
We (cid:12)rst show that, under the present action of H , the element (cid:14) is repre-
1 20
sentedbyaninnerderivationimplementedbytheabovementionedEisenstein
series of weight 4 and level 1, denoted here by ! . We then prove that no
4
inner perturbation of the action by a 1-cocycle can annihilate the Schwarzian
~
(cid:14) , and that the freedom one has in modifying the action of the Hopf alge-
20
bra H by a 1-cocycle exactly changes the restriction of the action of X on
1
modular forms and the value of ! as in the data used by Zagier [30] to de-
4
(cid:12)ne ‘canonical’ Rankin-Cohen algebras, where moreover the same ! appears
4
with a slightly di(cid:11)erent normalization.
4
We then show that the image under the canonical map from the Hopf cyclic
cohomology of H to the Hochschild cohomology of the algebras A((cid:0)) of the
1
Hopf 2-cocycle encoding the ‘transverse fundamental class’ gives the natu-
ral extension of the (cid:12)rst Rankin-Cohen bracket for modular forms to A((cid:0)).
Actually, in a sequel to this paper [15], we show that all the Rankin-Cohen
brackets have natural extensions to modular Hecke algebras, which moreover
are determined by a universal deformation formula based on H . The mere
1
existence ofthemodularHecke algebrasissu(cid:14)cient toprove infullgenerality
the universal associativity property.
While the transverse fundamental class generates the even component of the
periodic Hopf cyclic cohomology of H , the odd component is generated by
1
the Hopf cyclic version of the Godbillon-Vey class, namely (cid:14) . In the context
1
of the present action of H , this class gives rise to a ‘transverse’ 1-cocycle
1
on GL+(2;Q) with values in Eisenstein series of weight 2, measuring the
lack of GL+(2;Q)-invariance of the connection r corresponding to X. When
coupled with the obvious extension of the classical period cocycle, it yields
an analogue of the Bott-Thurston group 2-cocycle [7], that represents the
Euler class e 2 H2(SL(2;R); R) (cf. Theorem 16). By a theorem of Borel
d
and Yang [5], the restriction of this class to H2(SL(2;Q)(cid:14); R) is nontrivial
and generates H2(SL(2;Q)(cid:14); R).
Our last result (Theorem 17) gives an arithmetic representation for the ra-
tional Euler class e 2 H2(SL(2;Q)(cid:14); Q) in terms of generalized Dedekind
sums [16], [23]. The original Dedekind sums already arose in several di(cid:11)erent
topological settings, as amply illustrated by Atiyah [2] and also by Kirby and
Melvin [21].
InSection1weintroducethemodularHeckealgebraA((cid:0))foranycongruence
subgroup (cid:0) (cid:26) SL(2;Z). Its elements are form-valued Hecke operators; the
classical Hecke algebra is the subalgebra of elements of degree 0 for the
natural grading given by the weight.
Section 2 recalls the Hopf algebra H , in its original context of codimension 1
1
transverse geometry, and describes its most signi(cid:12)cant Hopf cyclic cocycles,
that correspond to the Godbillon-Vey class, to the Schwarzian derivative,
and to the transverse fundamental class.
In Section 3 we de(cid:12)ne and analyze in great detail the Hopf action of H on
1
the crossed product of modular forms by Hecke operators. We show that
the Schwarzian 1-cocycle gives an inner derivation. We relate the conjugates
5
by inner perturbations of the Hopf action of H to the data used by Za-
1
gier [30] to de(cid:12)ne canonical Rankin-Cohen algebras. We also show that the
Hopf cyclic 2-cocycle representing the transverse fundamental class provides
a natural extension of the (cid:12)rst Rankin-Cohen bracket to the crossed product
algebras. The section concludes with a precise formulation of the geometric
analogy with the case of Di(cid:11)(S1). In particular, we give a simple geometric
explanation for the above formula for the quadratic di(cid:11)erential $.
In Section 4 we introduce the ‘transverse’ 1-cocycle onGL+(2;Q)with values
in Eisenstein series of weight 2, as a partial analogue of the Godbillon-Vey
cocycle. We then complete the construction of its full analogue using the
‘period’ cocycle and prove that it represents the Euler class of SL(2;R).
Finally, we conclude by expressing the Euler class of SL(2;Q) in terms of
generalized Dedekind sums.
Appendix A recalls the de(cid:12)nition of cyclic cohomology for Hopf algebras,
whileAppendixBprovides thedetailsfortheinterpretationoftheGodbillon-
Vey class as a Hopf cyclic cohomological class.
We wish to express our gratitude to Don Zagier | this paper has greatly
bene(cid:12)ted from his course on modular forms at Coll(cid:18)ege de France [31] as well
as from his comments.
Contents
1 Modular Hecke Algebras 6
2 The Hopf algebra H and its Hopf cyclic classes 14
1
3 Hopf Symmetry of Modular Hecke Algebras 19
4 The transverse cocycle and the Euler class 37
1 Modular Hecke Algebras
Modular forms, regarded as lattice functions, may be viewed as natural co-
ordinates on the lattice line bundle over the moduli space of isomorphism
classes of elliptic curves over C. We recall that every elliptic curve is isomor-
phic { as a complex manifold { to a 1-dimensional complex torus E = C=(cid:3),
6
where (cid:3) is a lattice in C ; furthermore, two lattices (cid:3) and (cid:3) de(cid:12)ne iso-
0
morphic curves E ’ E i(cid:11) they are homothetic,
0
(cid:3) = (cid:21)(cid:3); for some (cid:21) 2 C :
0 (cid:2)
Since one can always represent a lattice (cid:3) in the form
!
(cid:3) = Z! +Z! ; Im 1 > 0;
1 2
!
2
uptothe obvious actionofthe group (cid:0)(1) = SL(2;Z), itfollows thatthe iso-
morphismclasses ofellipticcurves areparametrizedbythequotient (cid:0)(1)nH,
where H = fz 2 C;Im(z) > 0g is the upper half-plane. Thus, the set L of
all lattices in C de(cid:12)nes a line bundle
C ! L ! (cid:0)(1)nH;
(cid:2)
and the line bundle L 2 is canonically isomorphic to the (complex) 1-jet
(cid:0)
bundle of (cid:0)(1)nH.
Any lattice function which is homogeneous of weight 2k,
F((cid:21)(cid:3)) = (cid:21) 2kF((cid:3)); (cid:21) 2 C ;
(cid:0) (cid:2)
is automatically of the form
!
F(Z! +Z! ) = ! 2kf 1
1 2 2(cid:0) !
(cid:18) 2(cid:19)
with the function f satisfying the modularity property
fj (cid:13) = f ; 8(cid:13) 2 (cid:0)(1) = SL(2;Z); (1.1)
2k
where we used the standard ‘slash operator’ notation for the action of
G+(R) := GL+(2;R)
on functions on the upper half plane:
fj (cid:11)(z) = det((cid:11))k=2f((cid:11)(cid:1)z)j((cid:11);z) k; (1.2)
k (cid:0)
a b az +b
(cid:11) = 2 G+(R); (cid:11)(cid:1)z = and j((cid:11);z) = cz +d:
c d cz +d
(cid:18) (cid:19)
7
In particular, f(z + 1) = f(z); z 2 C, hence it induces a function on the
puncturedunitdisc f (q); q = e2(cid:25)iz. Suchafunctionf iscalledamodular
1
form of weight 2k if it is holomorphic on H and also at 1, that is f is
1
holomorphic at q = 0 ; f is called a cusp form if in addition f (0) = 0.
1
We denote by
M((cid:0)(1)) := (cid:6) M ((cid:0)(1)); resp. M0((cid:0)(1)) := (cid:6) M0 ((cid:0)(1));
(cid:8) 2k (cid:8) 2k
the algebra of modular (resp. cusp) forms.
A richer and more interesting picture emerges when lattices are replaced by
Q(cid:3) Q2
pairs ((cid:3);(cid:30)), with (cid:3) a lattice in C, and (cid:30) : (cid:0)(cid:24)! an isomorphism
(cid:3) Z2
which is ‘unimodular’ in a suitably de(cid:12)ned sense, cf. [24, x17]. The set L
A
of all such pairs forms again a line bundle, only this time over a projective
limit of Riemann surfaces,
C ! L ! H := lim (cid:0)(N)nH;
(cid:2) A A (cid:0)
N
where
a b a b 1 0
(cid:0)(N) = 2 SL(2;Z); (cid:17) modN :
c d c d 0 1
(cid:26)(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:27)
and the projective system is ordered by divisibility. The subscript A is justi-
(cid:12)ed by the adelic interpretation of the (cid:12)bration. Indeed, if one drops the
‘unimodularity’ condition on (cid:30), then the resulting space of pairs can be
canonically identi(cid:12)ed to the homogeneous space
GL(2;Q)nGL(2;A): (1.3)
The space L is the closure of GL(2;R) in this homogeneous space, and
A
its structure of principal bundle comes from the natural inclusion of the
multiplicative group C in GL(2;R). One can also describe H cf. loc. cit.
(cid:3) A
as a double quotient of the form,
H ’ GL(2;Q)nGL(2;A)0=K Z ; (1.4)
A
1 1
where K ’ S1, Z ’ GL(1;R) viewed as diagonal matrices in GL(2;R),
1 1
and the subgroup GL(2;A)0 is de(cid:12)ned by requiring that the determinant
belongs to Q (cid:2)R (cid:26) GL(1;A).
(cid:3) (cid:3)
8
Modular (resp. cusp) forms f of weight k with respect to any congruence
subgroup (cid:0) are de(cid:12)ned in a similar way as for (cid:0)(1), by requiring the holo-
morphicity (resp. cuspidality) of fj (cid:13) at 1, with respect to any local
k
parameter q1=m, for every (cid:13) 2 (cid:0)(1). One obtains, for each N (cid:21) 1, graded
algebras of forms of level N
M((cid:0)(N)) := (cid:6) M ((cid:0)(N)); resp. M0((cid:0)(N)) := (cid:6) M0((cid:0)(N));
(cid:8)k 1 k (cid:8)k 1 k
(cid:21) (cid:21)
which can be assembled together into algebras of modular (resp. cusp) forms
of all levels:
M := lim M((cid:0)(N)); resp. M0 := lim M0((cid:0)(N)):
(cid:0)! (cid:0)!
N N
!1 !1
The group
G+(Q) := GL+(2;Q);
acts ‘sideways’ on the tower de(cid:12)ning the projective limit H . Equivalently,
A
one can view G+(Q) as diagonally embedded in GL(2;A )0, with the latter
f
acting on H by right translations (cf. (1.4)). Using the fact that one can
A
factor any (cid:11) 2 G+(Q) in the form
a b
(cid:11) = (cid:13) (cid:1)(cid:12); (cid:13) 2 (cid:0)(1); (cid:12) 2 B+(Q) = 2 G+(Q) ; (1.5)
0 d
(cid:26)(cid:18) (cid:19) (cid:27)
it is easy to see that the action of G+(Q) preserves both holomorphicity
and cuspidality. Thus, to this action of G+(Q) corresponds a (cid:12)rst algebra of
‘noncommutative coordinates’ on the ‘ transverse space’ L =G+(Q), namely
A
the crossed-product algebra
A (cid:17) A := MoG+(Q);
G+(Q)
containing as an ideal the crossed product
A0 := M0 oG+(Q):
The elements of A are (cid:12)nite sums of symbols of the form
f U ; with f 2 M; (cid:13) 2 G+(Q);
(cid:13)(cid:3)
X
and with the product given by the rule
f U (cid:1)gU = (f (cid:1)gj(cid:11)) U ; (1.6)
(cid:11)(cid:3) (cid:12)(cid:3) (cid:12)(cid:3)(cid:11)
9
where the ‘slash operation’ is relative to the weight of g as in (1.2).
On the other hand, to the right action of GL(2;A )0 corresponds the induc-
f
tive limit of the following level N algebras, A(N) := A((cid:0)(N)), obtained as
crossed products of modular forms by Hecke operators at level N, which we
proceed now to describe.
More generally, for any congruence subgroup (cid:0) we shall construct an algebra
A((cid:0)), which contains as subalgebras both the algebra of (cid:0)-modular forms
M((cid:0)) as well as the Hecke ring at level (cid:0) (comp. [28, Chap. 3]), without
being in general generated by these subalgebras.
De(cid:12)nition. Let (cid:0) be a congruence subgroup. By a Hecke operator form of
level (cid:0) we mean a map
F : (cid:0)nG+(Q) ! M; (cid:0)(cid:11) 7! F 2 M;
(cid:11)
with (cid:12)nite support and satisfying the covariance condition
F = F j(cid:13); 8(cid:11) 2 G+(Q);(cid:13) 2 (cid:0): (1.7)
(cid:11)(cid:13) (cid:11)
A Hecke operator form F of level (cid:0) is called cuspidal if
F 2 M0; 8(cid:11) 2 G+(Q): (1.8)
(cid:11)
Thus, by de(cid:12)nition, F is determined by the (nonzero) values taken on a
(cid:12)nite number of inequivalent representatives f(cid:11) ;:::;(cid:11) g for double cosets
1 r
in (cid:0)nG+(Q)=(cid:0). It should be noted though that these values F 2 M are
(cid:11)i
notrequired tobemodularformsoflevel(cid:0). Mapsfrom(cid:0)nG+(Q)=(cid:0)toM((cid:0))
trivially ful(cid:12)ll equation (1.7), but they do not exhaust all its solutions. In
fact, one has:
Lemma 1. Let f 2 M be any modular form, (cid:0) a congruence subgroup.
There exists a Hecke operator form F of level (cid:0) such that F = f if and
(cid:11)
only if
fj(cid:13) = f ; 8(cid:13) 2 (cid:0) \(cid:11) 1(cid:0)(cid:11):
(cid:0)
10