( files chap2 to chap

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1 The basic objects in the cohomology theory of arithmetic groups Günter Harder This is an exposition of the basic notions and concepts which are needed to build up the cohomology theory of arithmetic groups We are dealing with objects which have a certain degree of complexity and in this text I give some explanation of the background needed to understand the the definitions and the general structural elements of these objects A complete discussion can be found on my home page 6 (wwwmathuni-bonnde/people/harder/manuscripts/buch/), files chap2 to chap A linear algebraic group G/Q is a subgroup G Gl N, which is defined as the set of common zeroes of a set of polynomials in the matrix coefficients where in addition these polynomials have coefficients in Q A prominent example is given by the orthogonal group of a quadratic form f(x 1,, x n ) = n a i x 2 i i= where a i Q and all a i 0 (this is actually not necessary for the following) We look at all matrices a 11 a 1N α = a N1 a NN which leave this form invariant, ie f(αx) = f(x) for all vectors x = (x 1,, x n ) This defines a set of polynomial equations for the coefficient a ij of α Instead of taking a quadratic form which is the same as taking a symmetric bilinear form we could take an alternating bilinear form, x, y = x 1,, x 2n, y 1, y 2n = n (x 1 y i+n x i+n y i = f x, y i=1 By G(Q) we denote the group of elements α Gl n (Q) which satisfy the given equations An arithmetic subgroup Γ G(Q) is a subgroup of the type Γ = Gl N (Z) G(Q), we may also require some congruence conditions for the coefficients The most famous example of an arithmetic group is the group Γ = Sl 2 (Z) The arithmetic subgroup Γ is also a subgroup of G(R), this is the subgroup where the entries are in R The group G(R) contains a maximal compact subgroup 1

2 K G(R) we can form the quotient space X = G(R)/K, if G/Q is semisimple then X carries a Riemannian metric with negative curvature an its connected components are diffeomorphic to some R d The group Γ acts on X from the left, this action is properly discontinuous, this means that for any point x X there exist a neighborhood U x such that for any γ Γ we have γ(u x ) U x = or γ is in the stabilizer Γ x = {γ γx = x}, this stabilizer is always finite It can happen that the group Z Γ of elements which act trivially on X is non trivial, ie Γ Z = {γ γ Γ, γx = x for all x X} {e} Then it is sometimes useful to replace Γ by Γ/Γ Z A point x X is called a fixed point if Γ x Γ Z Starting from the data Γ, X we form the quotient spaces Γ\X, these are so called locally symmetric spaces We consider the projection π : X Γ\X A point x Γ\X is called a fixed point if the points x π 1 ( x) are fixed points If our arithmetic group is Sl 2 (Z) then the symmetric space is the upper half plane H = {z C I(z) > 0}, we have the classical fundamental domain ) in the upper half plane (first picture) and after the necessary identifications the resulting quotient space is given by the second picture ( ) 1 1 (The two vertical lines are identified because T = Sl (Z) and T acts ( ) 0 1 by z z + 1, and on the bottom arc the matrix S Sl (Z) induces the reflection at the imaginary axis)

3 But we need more ingredients, we also consider Γ modules M, especially we are interested in Γ modules which come from rational representations of the given algebraic group G/Q If for instance our group is Sl 2 then a nice Γ module is the free Z-module of homogenous polynomials of degree n in two variables X, Y with coefficients in Z ν=n M n = { a ν X ν Y n ν a ν Z} ν=0 Such a Γ modules provides a sheaf of abelian groups M on Γ\X By definition a sheaf on Γ\X with values in the sheaf of abelian of abelian groups is a rule that 3

4 attaches to any open subset V Γ\X an abelian group M where this rule satisfies certain compatibilities with respect to restriction to smaller sets In our case we define M(V ) = {f : π 1 : V M f is locally constant, f(γu) = γf(u)} For any such such sheaf one can define the sheaf cohomology groups H q (Γ\X, M); q = 0, dim(x) In general the quotient spaces are not compact, but they have finite volume They admit a natural compactification, the so called Borel-Serre compactification i : Γ\X Γ\ X, then (Γ\ X) is the boundary Γ\ X \ Γ\X The sheaves extend to this compactification and we have an isomorphism i H q (Γ\X, M) H q (Γ\ X, M) We also have the notion of the the cohomology with compact supports H q c (Γ\X, M) and we get the fundamental long exact sequence H q 1 ( (Γ\X), M) H q c (Γ\X, M) H q (Γ\ X, M) H q ( (Γ\X), M) By a theorem of Raghunathan all the cohomology groups are finitely generated abelian groups provided the module M is a finitely generated abelian group If M is a finite dimensional Q vector space the same theorem says that the cohomology groups are finite dimensional Q vector spaces The last ingredient are the Hecke operators To any α G(Q) we can define a collection of endomorphisms T (α, u α ) : H (Γ\X, M) H (Γ\X, M), these endomorphisms also act on the cohomology with compact supports and the cohomology of the boundary, they are compatible with the long exact sequence Main objects of Study: Our main objects of interest are these cohomology groups as modules under the Hecke algebra H, this is the algebra generated by these endomorphisms I want to say a few words concerning the structure of the Hecke algebra H Remember that Γ G(Q) was the subgroup in which the matrix coefficients were integers, the matrix entries of α G(Q) have denominators Let us now fix a prime and consider only α for which the denominator is a power of our given prime p Then this yields a subalgebra H p, the local algebra at p Then we have the following a) The Hecke algebras H p and H p commute if p p b) For almost all primes p the Hecke algebra H p is commutative c) In a certain sense we have H = p H p 4

5 Finally I want to formulate a first fundamental theorem concerning the structure of the cohomology under the Hecke algebra We assume that our sheaf M is obtained from a rational representation of the algebraic group (like in the example above for Sl 2 ) We define the inner cohomology H q! (Γ\X, M) as the image H q! (Γ\X, M) = Image(H q c (Γ\X, M) H q (Γ\X, M)), this are finite dimensional Q vector spaces Then the theorem says Theorem:The action of the Hecke algebra H on the inner cohomology H q! (Γ\X, M) is semi simple, ie any Hecke-invariant submodule has a complement which is also Hecke invariant The proof of this theorem requires some analysis, we extend the sheaves to sheaves over the complex numbers, ie we replace M which is a sheaf of Q vector spaces by M C Then we apply Hodge theoretic methods to study H q! (Γ\X, M C), this means we apply some functional analysis and the theory of linear elliptic operators This allows to introduce a hermtian metric on the cohomology groups H q! (Γ\X, M C) for which the Hecke algebra is self adjoint and then the theorem follows In the classical case that Γ = Sl 2 (Z) We construct the Hecke operators but here we allow that α Gl 2 (Q) Especially we have Hecke operators T p = T ( ) p 0 0 1, u ( p ) where the second component u ( ) is given by the canonical choice (Compare p chap II 221 Then the Hecke-algebra is commutative and H p = Z[T p ] Here we assume that n is even because for n odd the sheaf is trivial Then we can apply our theorem and the commutativity of the Hecke algebra together with the semi simplicity yield a decomposition into isotypical components H 1! (Sl 2 (Z)\H, Mn ) = H 1! (Sl 2 (Z)\H, Mn )(Π f ), where the H 1! (Sl 2(Z)\H, Mn )(Π f ) are Hecke-invariant sub modules which have the following property: a) If I(Π f ) is the annihilator of the submodule H 1! (Sl 2(Z)\H, Mn )(Π f ), ie I(Π f ) = {T H T H 1! (Sl 2(Z)\H, Mn )(Π f ) = 0} then H Q/I(Π f ) Q = K(Π f ) is a field We denote the map H H/I(Π f ) also denoted by Π f b) Then H 1! (Sl 2(Z)\H, Mn )(Π f ) is a K(Π f ) vector space and this vector space is of dimension two c) We still have the action of the complex conjugation on the space Sl 2 (Z)\H, it induces an involution c on the cohomology which commutes with the action of the Hecke algebra Under this involution the space H 1! (Sl 2(Z)\H, Mn )(Π f ) decomposes into two one dimensional K(Π f ) vector spaces H 1! (Sl 2 (Z)\H, Mn )(Π f ) = H 1! (Sl 2 (Z)\H, Mn )(Π f ) c=1 H 1! (Sl 2 (Z)\H, Mn )(Π f ) c= 1 5

6 The first assertion is rather formal, the second assertion follows from the theory of automorphic forms We have to invoke the Eichler-Shimura isomorphism: If S n+2 (Sl 2 (Z)) is the space of holomorphic cusp forms of weight n + 2 then we have a canonical Hecke-invariant isomorphism Φ : S n+2 (Sl 2 (Z)) S n+2 (Sl 2 (Z) H 1! (Sl 2 (Z))\H, Mn C) Now the space of cusp forms decomposes into eigenspaces under the action of the Hecke-algebra: We have a finite number of homomorphisms λ : H C such that each such λ yields an eigenspace S n+2 (Sl 2 (Z))(λ) = {f S n+2 (Sl 2 (Z)) T p (f) = λ(t p )f} is of dimension one This one dimensional space is generated by a normalized cusp form f λ (z) = f(z) = q + a 2 q 2 + a 3 q 3 + a 4 q 4 = a n q n where q = e 2πiz and our space of cusp forms is the direct sum of these eigenspaces For a prime p we have the equality a p = λ(t p ), the eigenvalues are real (This is a classical theorem of Hecke) Now we get the Eichler-Shimura isomorphism as follows: We pick a Π f We consider the set of embeddings Σ = {σ : K(Π f ) C} it is known that all these embeddings factor through the field R of real numbers From any σ : K(Π f ) C, then we get a λ by the formula λ(t p ) = σ(π f (T p )), and this is just one of the λ which occur in the decomposition of S n+2 (Sl 2 (Z)) Then for a given Π f the numbers σ(π f (T p )) are algebraic integers, they are real and conjugate under the Galois group of Q/Q Now we can-again following Hecke- attach an L-function to any such λ = σ Π f, it is defined as the infinite product n=1 Λ(σ Π f, s) = Γ(s) 1 (2π) s 1 λ(t p )p s + p n+1 2s p Several things are known about this function: First of all we have the deep theorem of Deligne which says that for all n and all λ we have λ(p) 2p (n+1)/2 This implies that the infinite product converges in the half-plane R(s) > n and defines a holomorphic function in this half plane But we know more This function has an analytic continuation into the entire complex plane and satisfies a functional equation 6

7 Λ(σ Π f, s) = ( 1) n/2+1 Λ((σ Π f, n + 2 s) Finally we have a theorem of Manin and Shimura which says: We can define two arrays (the periods) Ω = {, Ω ± σ, } σ Σ R #Σ such that for any integer ν with 1 ν n + 1 we find a number Λ(Π f, ν) K(Π f ) such that σ((λ(π f, ν)) = L(σ Π f, ν) Ω ɛ(ν) σ where ɛ(ν) is the parity of ν If the class number of K(Π f ) is one then we can pin down the periods up to a pair of units ɛ ± O F, ie we have a procedure of choosing them such that for any two pairs Ω,±, Ω ± we find a pair of units ɛ ± such that Ω,± = {, σ(ɛ ± )Ω ± σ, } σ Σ For 2 ν n The numbers Λ(Π f, ν) K(Π f ) are algebraic integers, if ν = 1 or ν = n then the numbers (1 + p n+1 Π f (T p ))Λ(Π f, ν) K(Π f ) are algebraic integers If the class number is not one then we can find a finite set S of primes which is disjoint from a given prime p such that the ring R = O F,S has class number one Then we can choose our periods in such a way that they are unique up to an element in R Especially we know that the number ord p ( L(σ Π f, ν) ) Ω ɛ(ν) σ is well defined It is a general idea that the arithmetic of these numbers Λ(Π f, ν), ie their decomposition into prime ideals contains some arithmetic information This hope is supported by some experiments Actually the proof of the theorem of Manin and Shimura is not so difficult Let f be the cusp form corresponding to σ Let us assume for simplicity that our field K(Π f ) = Q, the we get only one modular form f and its coefficients a n Q We have to write down the classical integral representation of the L-function as a Mellin-transform Λ(f, s) = 0 f(iy)y s dy y Via the Eichler-Shimura isomorphism the modular form f and its complex conjugate provide two cohomology classes ω ± = f ± f H 1! (Sl 2 (Z)\H, Mn C)(±), where we notice that the complex conjugation acts on the cohomology and decomposes it into a plus and a minus eigenspace We also have this decomposition into a plus and a minus eigen-module for the integral cohomology 7

8 H 1! (Sl 2 (Z)\H, Mn ) H 1! (Sl 2 (Z)\H, Mn )(+) H 1! (Sl 2 (Z)\H, Mn )( ) where the right hand side may have an index which is 1 or 2 Then we find two numbers Ω ± (f) such that ω ± Ω ± is a generator of H 1! (Sl 2 (Z)\H, Mn )(±) Now we can consider the homology groups H 1 (Sl 2 (Z)\H, M n ) they are defined as the homology of the complex of singular chains with coefficients in the cosheaf M n One such chain is simply the line from 0 to i along the imaginary axis We can multiply this 1-chain with a coefficient X ν Y n ν and we get a relative homology class [[0, i ] X ν Y n ν ] H 1 (Sl 2 (Z)\H, (Sl 2 (Z)\H)), M n ) If now ν 0, n then it is easy to see that this class lifts to a class in H 1 (Sl 2 (Z)\H, M n ) and by interpretation we get that < [[0, i ] X ν Y n ν ], ω ± Ω ɛ(ν) >= Λ(f, n + 1 ν) Z If ν = n, 0 then we have to modify the relative cycle by an application of a Hecke-operator if we want to bound it at infinity, this introduces the factor in front I add a table with a few modular forms Then we have the modular cusp form of weight 12 (z) = q 24q q q q q q q 8 of weight 22 f 22 (z) = q 288q q q q q 6 and two forms of weight 24 f 24 (z) = q + ( )q 2 + ( )q 3 f 24(z) = q + ( )q 2 + ( )q 3 Here the field K(Π f ) is the field K = Q[ ) = Q[α] where the only thing we know about α is that α 2 = So the two forms are indistinguishable For the modular form f = f 22 we get the following list of special values 8

9 {Λ(f, ν) Ω } ν=21,19,,11 = { , , , , 5 3 7, 0} and for the even case {Λ(f, ν) Ω + } ν=20,18,,12 = { , , , , 2 3 7} In the Cahit Arf lecture I will explain what the arithmetic implications of the prime number 41 occurring in the value Λ(f,14) Ω + should be, for further details I refer to my article in The of modular forms which appeared at Springer It is already difficult to make the list for the modular form of weight 24 because I do not know what the class number of Q( ) is and I have difficult to fix the periods 9