Invariants in relative cohomology and compact support cohomology of the quotient

Question

Let $\cal H$ be the Poincare upper half-plane and $\overline {\cal H}$ the union of $\cal H$ with the set of cusps $\bf P^1 (\bf Q)$, provided with its usual topology. Let $\Gamma$ a congruence subgroup acting freely on $\cal H$, $V$ an abelian group with $\Gamma$-action, and $\tilde V$ the associated local system on the quotient $\cal H / \Gamma$.

Ash and Stevens claim that there exists a natural isomorphism $$H^1(\overline {\cal H},{\bf P^1}({\bf Q}),V)^\Gamma = H^1_c({\cal H} / \Gamma, \tilde V).$$ This is in their paper in Duke, vol. 53, no 3, 1986, "MODULAR FORMS IN CHARACTERISTIC $\ell$ AND SPECIAL VALUES OF THEIR L-FUNCTIONS", page 862. They don't give any justification or proof.

Can someone explain this isomorphism, or point out the general results in algebraic topology with which one could prove it?

Answer 1

In order to understand the isomorphism, I would translate everything to cohomology of sheaves and then use both Grothendieck's spectral sequences that converge to the same equivariant cohomology groups of sheaves of abelian groups on $\overline{\mathcal H}$ having a $\Gamma$-action.

A relative cohomology group $H^\star(X,A,V)$ of a pair $(X,A)$ with values in an abelian group $V$, or more generally, with values in a sheaf of abelian groups is nothing but the cohomology group $H^\star(X,j_!V)$, where $j$ is the inclusion of the open subset $U=X\setminus A$ into $X$. If $V$ is an abelian group, $j_!V$ is the extension-by-$0$ to $X$ of the constant sheaf $V$ on $U$.

In your particular situation, $X=\overline{\mathcal H}$, $A=\mathbf P^1(\mathbf Q)$ and $U=\mathcal H$, and we have $$ H^1(\overline{\mathcal H},\mathbf P^1(\mathbf Q),V)=H^1(\overline{\mathcal H},j_!V). $$ By naturality of this isomorphism, it even holds $\Gamma$-equivariantly so that, in particular, $$ H^1(\overline{\mathcal H},\mathbf P^1(\mathbf Q),V)^\Gamma=H^1(\overline{\mathcal H},j_!V)^\Gamma. $$

Now, $j_!V$ is a $\Gamma$-sheaf on $\overline{\mathcal H}$, meaning a sheaf of abelian groups endowed with a $\Gamma$-action, lying over the $\Gamma$-action on $\overline{\mathcal H}$. Grothendieck's second spectral sequence (see his famous Tohoku article, part 2, Théorème 5.2.1, p. 200) is denoted by $I\!I_r^{pq}$. It's second page is group cohomology of the cohomology groups of $j_!V$: $$ I\!I_2^{pq}=H^p(\Gamma,H^q(\overline{\mathcal H},j_!V)) $$ in your situation. It converges to the equivariant cohomology groups $$ H^{p+q}(\overline{\mathcal H},\Gamma,j_!V). $$ Note that the whole first row $I\!I_2^{p0}$ is zero since the only global section of $j_!V$ is the trivial one. Hence, one gets $$ H^1(\overline{\mathcal H},\Gamma,j_!V)=I\!I_2^{01}=H^1(\overline{\mathcal H},j_!V)^\Gamma. $$

As for Grothendieck's first spectral sequence, since the quotient map $$ f\colon \overline{\mathcal H}\rightarrow \overline{\mathcal H}/\Gamma $$ has discrete fibers, its second page is $$ I_2^{pq}=H^p(\overline{\mathcal H}/\Gamma, H^q(\Gamma,f_\star j_!V)) $$ (see loc. cit., Proposition 5.2.2, p. 201). Here, $f_\star j_!V$ is the push-forward of the sheaf $j_!V$ to the quotient $\overline{\mathcal H}/\Gamma$. It is a $\Gamma$-sheaf on the quotient, when the quotient is considered with the trivial $\Gamma$-action. The group $I_2^{pq}$ is the cohomology group on the quotient with values in the group cohomology sheaf $$ H^q(\Gamma,f_\star j_!V). $$ This time, the whole first column $I_2^{0q}$ is zero. Hence, $$ H^1(\overline{\mathcal H},\Gamma,j_!V)=I_2^{10}=H^1(\overline{\mathcal H}/\Gamma, (f_\star j_!V)^\Gamma). $$ The sheaf $(f_\star j_!V)^\Gamma$ is your local system $\tilde V$, but considered on the open subset $\mathcal H/\Gamma$, and extended-by-$0$ as a sheaf on the whole quotient $\overline{\mathcal H}/\Gamma$. Since this quotient is compact, one has $$ H^1(\overline{\mathcal H}/\Gamma, (f_\star j_!V)^\Gamma)=H_c^1(\mathcal H/\Gamma,\tilde V). $$