In Diamond, Shurman: A First Course in Modular Forms on p. 77, the meromorphic differentials on an open set $V\subset \mathbb C$ of degree $n$ are defined as $$\Omega^{\otimes n}(V)=\lbrace f(q)(dq)^n\colon f\text{ meromorphic on }V\rbrace,$$ where $q$ is a variable on $V$. What is the rigorous definition of the RHS above? Can it be defined as a tensor product of something?
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1$\begingroup$ It's a section of the $n$-th symmetric power of the bundle of meromorphic differentials. $\endgroup$ – Angina Seng Jun 6 '20 at 20:30
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The notation $\Omega$ is used for the cotangent bundle of a complex manifold. Its sections are known as holomorphic differential forms. In the case of $V \subseteq \mathbb{C}$, the tangent bundle is the trivial line bundle. Therefore, $\Omega = \Omega^1$, the differential $1$-forms, and given a local coordinate $q$, the differential $1$-form $dq$, which is the dual of the vector field (section for the tangent bundle) $\frac{\partial}{\partial q}$, is a basis for this $1$-dimensional space.
As with any bundle, one may consider its meromorphic sections, i.e. elements $\omega \in \Omega(U)$ where $U$ is an open subset such that $V \setminus U$ consists of isolated points, and $\omega$ has poles in $V \setminus U$. Note that in this case this is simply $\Omega^{(1)}(V) = \{ f(q) dq \}$ where $f(q)$ is a meromorphic function.
Finally, we can tensor bundles. One can think of $\Omega^{\otimes n}$ as the $n$-power tensor bundle of $\Omega^{(1)}$, or as multilinear maps on $n$-tuples of vector fields. However, in this degenerate case, this is again a trivial line bundle, spanned by the element $(dq)^{\otimes n}$, which is denoted in the book as $(dq)^n$. Its meromorphic sections are once more simply of the form $f(q) (dq)^{\otimes n}$, where $f$ is a meromorphic function.
