According to wikipedia
"A modular form of weight $k$ for the modular group $SL(2, \mathbf Z)$ is a complex-valued function $f$ on the upper half-plane $H = {z ∈ C, Im(z) > 0}$, satisfying the following three conditions:
(1) $f(z)$ is a holomorphic function on $H$.
(2) $$f(-1/z) = z^k f(z)\tag{2A},$$ and $$f(z+1) = f(z)\tag{2B}$$.
(3) $f(z)$ is required to be holomorphic as z → i∞.
Question: If only condition (2B) is not satisfied,but (1),(2A),(3) are still satisfied, what is the name of this form/function $f(z)$?
I suppose that $B=2$ ....
The space of holomorphic function on the upper halfplane which are holomorphic at $\infty$ and satisfying $f(z+1)=f(z)$ is isomorphic to the space of holomorphic function on the open unit disc $D$.
This is so because there's a biholomorphic isomorphism ${\cal H}/{\Bbb Z}\simeq D\setminus\{0\}$ and the condition for $f$ to be holomorphic at $\infty$ is exactly that assuring that the corresponding function extends holomorphically to $0$.
If, instead, $f(z+1)=f(z)$ is the dropped relation, note that $z\mapsto-\frac1z$ is an involution of $\cal H$ with $z=i$ as unique fixed point. The shape of the functional equation tells immediately that $k$ must be even (for $f$ to be non-trivial) and the holomorphic functions are in one-to-one correspondence with the holomorphic functions on the quotient space $\cal H/\sim$ with the extra condition that they must vanish at $\bar\imath$ if $k\equiv2\bmod4$.