Does this pairing-based signature scheme work?
Your proposed signature scheme falls to universal forgeries under a known message attack (UF-KMA).

The adversary $\mathcal{A}$ receives the public key $(A,P)$, a single message signature pair $(M,C)$ and the challenge message $M^*$.
It outputs $$C' := \frac{H(M)}{H(M^*)} \cdot C.$$

The adversary described above is succesful with probability $1$.
To see this, consider that $C$ is by definition
$$C = \frac{p}{a\cdot H(M)} \cdot G_2.$$
We now have 
\begin{align}
C' &amp;= \frac{H(M)}{H(M^*)} \cdot C\\
&amp;= \frac{H(M)}{H(M^*)}\cdot\frac{p}{a\cdot H(M)} \cdot G_2\\
&amp;=\frac{p\cdot H(M)}{a\cdot H(M) H(M^*)} \cdot G_2\\
&amp;=\frac{p}{a\cdot H(M^*)} \cdot G_2,
\end{align}
which is exactly the signature of $M^*$ and will therefore be accepted by the verification equation.