@SubhajitJana Ah yes, you are one-hundred percent correct. This is more a function of my ignorance than of your misunderstanding, although I do think I knew that deep down. :) Thanks! I will edit.

Dear Ariyan, thank you so much for your kind answer! I just sent you an email with some questions.

Do you have any intuition about why the argument I said (which is supposed to be similar to your addendum's argument) extends to the positive characteristic case and the method you described cannot be adapted to work fail for higher dimensional examples? Namely, is there an intuition about why your analytic result on the Hermitian symmetric domain $\mathfrak{h}_n$ cannot be extended to an algebraic result about the locally symmetric space $\mathfrak{h}_n/\Gamma(m)$? Thanks so much!

but the method in your addendum cannot apply since there are non-isotrivial abelian surfaces over $\mathbb{A}^1_{\overline{\mathbb{F}_p}}$. Your argument essentially relies on the complex differential geometry of $\mathfrak{h}_n$ opposed to, say, the algebraic properties of the quotient $\mathfrak{h}_n/\text{Sp}_{2n}(\mathbb{Z})$ (or further quotients of that).

Of course, this corresponds to a map $\mathbb{A}^1_\mathbb{C}\to Y(N)$ which extends to a map $\mathbb{P}^1_\mathbb{C}\to X(N)$. But, for $N\gg 0$ we know that $g(X(N))>0$ and so any map $\mathbb{P}^1_\mathbb{C}\to X(N)$ must be constant which gives the isotriviality. Now, this argument is, according to ACL, spiritually the same as your addendum argument. But, there is one big difference that I can see. The method I described above extends to show, for example, that all elliptic schemes over $\mathbb{A}^1_{\overline{\mathbb{F}_p}}$ (or even $\mathbb{A}^1_{\mathbb{F}_p}$) are isotrivial.

Hey Donu, I was wondering if you could offer some insight into the following question concerning your addendum proof. Below in the comments of ACL's answer I give a proof for why there can be no non-isotrivial elliptic schemes over $\mathscr{E}/\mathbb{A}^1_\mathbb{C}$. Essentially $\mathscr{E}[N]$ must be constant for all $N$ (since $\mathbb{A}^1_\mathbb{C}$ is (étale) simply connected) and thus for some choice of $\alpha:(\mathbb{Z}/N\mathbb{Z})^2\mid_{/\mathbb{A}^1_\mathbb{C}}\cong \mathscr{E}[N]$ we get that the pair $(\mathscr{E},\alpha)$ defines a point of $Y(N)$.

Thanks so much! I'm going to accept Donu's answer because it answers all three of my questions, but I greatly appreciate your help.

Thanks Donu! This makes me very happy. One question: is Q2 obvious without using the cited fact that a $\mathbb{Q}$-VHS is constant? Namely, it's clear that this+Scmid's result imply Q2, but am I missing some obvious way of avoiding this result of Steenbrink and Peters?

@PiotrAchinger Hey Piotr! Thanks, I'll look into this. I'm having trouble locating the original paper. Do you have a clue what the title is? Thanks!

Also, in this paper it's not clear to me whether the following is true. In the case of elliptic schemes the following works: choose a trivilization $\alpha:\mathscr{E}[n]\xrightarrow{\approx}(\mathbb{Z}/N\mathbb{Z})^2$. This then defines a map $\mathbb{A}^1_\mathbb{C}\to Y(N)$ which extends to a map $\mathbb{P}^1_\mathbb{C}\to X(N)$. For $N\gg 0$ the genus of $X(N)$ is positive and so this map must factor through a point—thus the family is isotrivial. Do you know if there is a way to proceed using the geometry of $\mathcal{A}_{g,1}$ (or $\mathcal{A}_{g,n}$, or its spaces with level structure)?

Hey ACL, thanks for the information! Do you have a belief that $\mathscr{A}$ should be isogenous to a Jacobian (as in the case of fields)? And is your statement 'I don't know whether there is a reference in the literature' mean "I think it's true" or "It is true—no one's bothered to type it up yet"? Also, do you have any opinion about the 'proof'I described about above?