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Let $X_0$ be an irreducible smooth projective curve defined over $\overline{\mathbb Q}$ and $f_0 : X_0 \rightarrow \mathbb{P}^1_{\overline{\mathbb Q}}$ a nonconstant morphism whose branch locus is contained in the subset $\{0,1, \infty\} \subset \mathbb{P}^1_{\overline{\mathbb Q}}$. For any vector bundle $E$ on $X = X_0\times_{{\rm Spec}\,\overline{\mathbb Q}} {\rm Spec} \mathbb{C}$, consider the direct image $f_*E$ on $\mathbb{P}^1_{\mathbb C}$, where $f= (f_0)_{\mathbb C}$. It decomposes into a direct sum of line bundles and also it has a natural parabolic structure. We prove that $E$ is the base change, to $\mathbb C$, of a vector bundle on $X_0$ if and only if there is an isomorphism $f_*E \stackrel{\sim}{\rightarrow} \bigoplus_{i=1}^r {\mathcal O}_{{\mathbb P}^1_{\mathbb C}}(m_i)$, where $r = {\rm rank}(f_*E)$, that takes the parabolic structure on $f_*E$ to a parabolic structure on $\bigoplus_{i=1}^r {\mathcal O}_{{\mathbb P}^1_{\mathbb C}}(m_i)$ defined over $\overline{\mathbb Q}$., Comment: Final version; to appear in Michigan Mathematical Journal

Let $C$ be a smooth projective curve defined over the field of complex numbers. Let $E$ be a vector bundle on $C$, and fix an integer $d\geqslant 1$. Let $\mc Q:={\rm Quot}(E,d)$ be the Quot Scheme which parameterizes all torsion quotients of $E$ of degree $d$. In this article, we survey some recent results on various invariants of $\mc Q$., Comment: Final version; to appear in ICTS Conference proceedings. arXiv admin note: text overlap with arXiv:2203.13150, arXiv:2006.16666