Hi,

Fix $N > 3$ and consider the modular curve $X(N)$ parametrizing elliptic curves with full level N structures. Let $\pi : E(N)\to X(N)$ be the universal elliptic curve. Then $V=R^1\pi_*\mathbf Q$ defines a local system on $X(N)$. Define $H^1_c = H^1_c(X(N),Sym^kV)$ and $H^1=H^1(X(N),Sym^kV)$ for $k>=0$. Parabolic cohomology is defined as $H^1_p = image H^1_c\to H^1$. Then there is a direct sum decomposition

$$ H^1 = H^1_p \oplus H^1_e$$

where $H^1_e$ is called the Eisenstein part of cohomology.

My question is: how does one prove that $H^1_p$ is in fact a direct summand of $H^1$? And is there any description of $H^1_e$ in terms of the geometric objects $X(N)$, $E(N)$?

Thanks!