Let $V_{k}$ denote the complex representation of $\mathrm{GL}(2)$ given by $\mathrm{Sym}^k(V)$, where $V$ is the defining 2-dimensional representation. Assume that $k$ is even. I would like to compute the cohomology group $$ H^1(\mathrm{GL}(2,\mathbf Z),V_{k}). $$ By the Lyndon-Hochschild-Serre spectral sequence, this is the same as the $\mathbf Z/2$-invariants in $H^1(\mathrm{SL}(2,\mathbf Z),V_k)$. The latter cohomology group is a direct sum of three terms, each of which corresponds to a space of modular forms of weight $k+2$: one to holomorphic cusp forms, one to antiholomorphic cusp forms, and one to Eisenstein series. I think that the $\mathbf Z/2$-invariants consist exactly of the holomorphic cusp forms, but I can't seem to figure out why. Can someone give an argument and/or a reference that explains this, preferably in geometric terms?

Thanks to Joël's great answer I think I understand what's going on. Let me try to make really explicit where my confusion came from. If you write the Eichler-Shimura isomorphism $$ H^1_{\mathrm{cusp}}(\mathrm{SL}(2,\mathbf Z),V_k) \cong S_{k+2} \oplus \overline S_{k+2}$$ then it's not true as I thought (and which led to nonsensical answers) that the subspace $S_{k+2}$ is $\mathbf Z/2$-invariant and $\overline S_{k+2}$ is anti-invariant. Instead the $\mathbf Z/2$-action interchanges these two subspaces according to the involution $$ f(z) \mapsto -f(-\overline z),$$ which maps holomorphic modular forms to antiholomorphic ones and vice versa. (I guess depending on how you've defined the Eichler-Shimura isomorphism there might be a sign difference here.) Still, there is of course a natural way to identify the invariant subspace with the space of holomorphic cusp forms.


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Let me try an answer. Instead of working with $H^1(SL(2,{\bf Z}),V_k)$, I'll work with a space which is naturally isomorphic to it, but more concrete, the space of modular symbols $Symb(V_k)$, defined as $Hom(\Delta^0,V_k)^\Gamma$. Here, $\Delta^0$ is the abelian group of divisors of degree $0$ on the projective rational line $\mathbb P^1(\bf Q)$, and $\Gamma = SL_2(\bf Z)$ for shortness. The isomorphism between those two spaces is standard, follows from the long exact sequence of cohomology attached to the pair (Poincaré half-plane with cusps $P^1(\bf Q)$ attached, $P^1(\bf Q)$), and commute with the action of Hecke operator on both spaces, in particular, what matters to us, with the action of the "Hecke operator at $\infty$" $\iota$ defined by the matrix $\left( \begin{matrix} 1 & 0 \\ 0 & -1\end{matrix}\right)$ (a generator of $GL_2({\bf Z})/SL_2({\bf Z})$). Also I see $V_k$ as the dual of the space of polynomial in one variable z of degree $\leq k$, with the natural homographic action of $\Gamma$. That's equivalent to your definition as a $Sym^k$.

Now we need to describe explicitly the relation you mention between modular forms and cohomology/modular symbols. Let us begin by cusp forms. In your post, the $\Gamma$-module $V_k$ is a complex vector space, but of course it is naturally defined over $\bf Z$, and I will call $V_k(\bf R)$ the version over $\bf R$. Let $S_{k+2}=S_{k+2}(\Gamma,\bf C)$ be the space of cusp forms. Then there is an injective $\bf R$-linear map $$S_{k+2} \rightarrow Symb(V_k({\bf R})),\ \ f \mapsto \phi_f,$$ where $\phi_f$ is the modular symbol defined by $$\phi_f(D)=(P \mapsto \Re( \int_D f(z) P(z) dz)).$$ Here, for $D=[a] - [b]$, $\int_D$ means the integral along any geodesic in the Poincaré half plane relying the cusp $a$ to the cusp $b$, for more general $D \in \Delta^0$, $\int_D$ is defined by linearity, and we see the right hand side is a linear form on the space of polynomials $P$ of degree $\leq k$ with real coefficients, hence an element of $V_k(\bf R)$ as it should be.

Now if $f = \sum a_n q^n \in S_{k+2}$, let $f'=\sum \bar a_n q^n \in S_{k+2}$. The $\bf R$-linear involution $f \mapsto f'$ is of course the complex conjugation on $S_{k+2}$ associated with its canonical real structure given by the forms defined over $\bf R$ (= with real coefficients). Note that $\overline{f(z)} = \sum \bar a_n \overline{e^{2 i \pi n z}} = \sum \bar a_n e^{-2 i \pi n \bar z}=f'(-\bar z)$. If $D=[\infty]-[a] \in \Delta^0$, $a \in \bf Q$, one has $\iota(D)=[\infty]-[-a]$ one has: $$\Re \left( \int_D f(z) P(z) dz \right) = \Re \left( \int_0^\infty f(a+iy) P(a+iy) i dy \right) \\ = \Re \left( \overline{ \int_0^\infty f(a+iy) P(a+iy) i dy } \right) \\ = - \Re \left( \int_0^\infty f'(- a+iy) P(a-iy) i dy \right) \\ = -\Re \int_{\iota D} f'(z) P(-z) dz, $$ hence $$\phi_{f'} = - \iota (\phi(f)),$$ that is the $\iota$ involution on modular symbols corresponds to minus the complex conjugation on modular cusp forms.

Now we complexify this real linear map, getting an injective map $$S_{k+2}(\Gamma,{\bf C}) \otimes_{\bf R} {\bf C} \hookrightarrow Symb(V_k({\bf C}),$$ where the left hand side can be identified with two copies of $S_{k+2}(\Gamma,{\bf C})$ and clearly from the above and easy linear algebra, one copy is identified with the $+1$-eigenspace, an other with $-1$ eigenspace for $\iota$ on the RHS. So this justifies your claim that the subspace invariant by $\iota$ in $Symb(V_k)$ contains one copy of the holomorphic modular cusp forms.

There is also the question of Eisenstein series. Working in level 1 as you do, there is only one of them for each k, and the question if whether the modular symbol corresponding to it is of eigenvalue $+1$ or $-1$ for $\iota$. This implies unfortunately that to have it right we need to be very careful in choosing a consistent system of conventions, which is a little bit too hard for me. If I interpret correctly my own computations with Samit Dasgupta however in our paper THE p-ADIC L-FUNCTIONS OF EVIL EISENSTEIN SERIES (Prop. 2.6), to appear in compositio, I think this sign is +1, which would mean that the Eisenstein series is also in the invariant subspace of $Symb(V_k)$, in other words also appears in $H^1(GL_2({\bf Z}),V_k)$. but maybe I'm wrong. What makes you think the latter is constituted of only cuspidal nodular forms?

  • $\begingroup$ Thanks a lot! I don't completely understand yet, but this looks very good. Is there a canonical reference? For my application I actually only need to understand the cusp forms. The reason I thought the Eisenstein series would be anti-invariant is the second-to-last paragraph of p.12 of arxiv.org/abs/math/0611847 (which I found when googling this problem). $\endgroup$ Sep 28, 2014 at 5:03

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