I'm reading the proof of algebraicity theorem for L-functions associated with cusp forms from Hida's book.

In this proof,the author related L-function's values with a differential form on the modular curve by integral.And he give a interpretation of this integral by a morphism of cohomology groups.

He said,for a modular curve Y,we can compact Y by adding a circle $S^1$ on each cusps(the author call it the Borel-Serre compactification), call this compactification $X^*$. Then for each $r \in Q$, the vertical line $C^r$ connecting r and the cusp $\infty$ is a relative cycle in $H_1(X^*,\partial X^*;Z)$, where $\partial X^*=\bigcup S^1$.Identifying $C^r$ with $R_+=\{x\in R|x >0 $},then there will be a natural isomorphim induce by $R_+ \rightarrow X^*:$ $$ Int_r:H_c^1(Y,\mathfrak{L}(n;A)) \longrightarrow H_c^1(R_+,\mathfrak{L}(n;A))=L(n;A)$$ Where $\mathfrak{L}(n;A)$ is the local constant sheaf from L(n;A),the homogeneous polynomials of degree n with X and Y.

My questions are:

1.What are the elements in $H_c^1(Y,\mathfrak{L}(n;$A$))$ looks like?I know
that when $K$ is a group ,the cohomology group $H^1(X,K)$ consists of homomorphisms between the 1-cocycle classes and K.

2 How $Int_r$ induced from $R_+ \rightarrow X^*$ ? Can we write down how $Int_r$ works? For $R_+$ is a subset of $X^*$,I know there is a homomorphism $$ \varphi: H_c^1(X^*,\mathfrak{L}(n;A)) \longrightarrow H_c^1(R_+,\mathfrak{L}(n;A))$$ which seems like restrict elements in $H_c^1(X^*,\mathfrak{L}(n;A))$ from $X^*$ to $R_+$.

The materials come from Hida's book $Elementary\ theory\ of\ L$-$functions\ and\ Eisenstein\ series$, chapter 6,section 4.

Thanks for your help.


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