In this paper we extend to algebraic extensions of local and global fields and their completions some classical results due to Borel-Serre, Tits, Conrad, Douai, Kneser and Sansuc concerning the finiteness, the surjectivity of maps between Galois cohomology groups and the obstruction to weak approximation and some related results of connected linear algebraic groups.

We extend results of Knopp in [9] to the higher level case. In precise, we characterize a rational period function $q(z)$ for $\Gamma_{0}^{+}(2)$ of which poles lie only in $\mathbf{Q} \cup \{\infty\}$. We prove that the Mellin transform $\Phi_{F}(s)$ of an entire modular integral $F$ of weight $2k$ for such a rational period function $q(z)$ has an analytic continuation to the entire $s$-plane, except for possible simple poles at some rational integers, satisfies the functional equation $\Phi_{F}(2k-s) = (-1)^{k} 2^{s-k} \Phi_{F}(s)$, and is bounded on each “truncated strip” of the from $\sigma_{1}\leq \mathop{\mathrm{Re}} (s)\leq\sigma_{2}$ and $|{\mathop{\mathrm{Im}}}(s)|\geq t_{0}>0$. We also show that the converse is true. The case for $\Gamma_{0}^{+}(3)$ is addressed similarly.

In [Cho09], Choi studied congruences of coefficients (modulo $T^{q}-T$) for Drinfeld modular forms of level $\Gamma_{0}(T)$, trivial type and the linear relations between the initial coefficients of those. In this article, we generalize these results for level $\Gamma_{0}(T)$, arbitrary type.