That looks right to me

Sorry, I misspoke. I believe any totally ramified extension of $\mathbb{Q}_p$ has monogenic ring of integers over $\mathbb{Z}_p$, generated by any uniformizer. See here: math.stackexchange.com/questions/117973/…

Completing at $\mathfrak{p}$ means that any element of $K$ whose $\mathfrak{p}$-valuation is 1 maps to a uniformizer in the completion. $K_\mathfrak{p}$ is an unramified extension of $\mathbb{Q}_p$, so its ring of integers is monogenic over $\mathbb{Z}_p$. So I think $\mathfrak{p}^2 \mathbb{Z}_p = \mathfrak{q}^2$.

Also perhaps the projection formula to show that the pushforward plays nicely with the short exact sequence computing local sections: math.stackexchange.com/questions/344126/…

Yes, the cohomology is exactly the same because a finite Cech cover of $\mathbb{P}^n$ by open affines induces one of $X$, and this gives an isomorphism of chain complexes by the definition of the pushforward. Also you may need that $j$ is an affine map so the pushforward on coherent sheaves is an exact functor.

Another way to see this: $\mathbb{Q}$ is a prime field, so any isomorphism must be a map of extensions of $\mathbb{Q}$, and therefore maps the rings of integers of the respective fields onto each other. But $p$ is a unit in the ring of integers of $\mathbb{Q}_l$, whereas it is not a unit in the ring of integers of $\mathbb{Q}_p$.

If $f(z)$ can be made arbitrarily close to $|z|$, then shouldn’t compactness of the closed unit disc and continuity of $f$ guarantee that equality is attained somewhere?

This paper seems to address the question, and the upper bound indeed appears to be $3n - 2$. I didn’t read it in detail though. I assume you meant to require $n > 1$. ams.org/journals/proc/2003-131-02/S0002-9939-02-06476-6/…

I’m not a professional researcher, but I imagine it mostly comes down to reading and writing. Read a lot of books and papers, and when they state a proposition, before you read the proof, at least ask yourself “why should this be true?”, if not attempt to prove it yourself. I think most mathematicians only carry a few different hammers, and go out looking for nails. There’s only so many things one mind can find intuitive. Even Euler relied on his core expertise with formal algebra and generating functions.

I would appreciate if you posed the problem in terms of typical mathematical notation, rather than talking about cups of water.

Now you have changed its definition to read $||(P_k)||_{\infty;K}$ with $k$ as a subscript— but you are maximizing over $0 \le k \le m$ so it does not depend on $k$. Personally I find it confusing (and would just replace the entire symbol $||(\cdot)||$ with a function $f(P, K)$, since as written it looks very similar to the norm $||\cdot||$).

It looks like in several places you write $m$ where you mean to write $k$. As written the definition of $P_k$ does not depend on $k$. It would also be nice if instead of $||(P_m)||_{\infty;K}$ you just wrote $||(P)||_{\infty;k}$, since the former makes it look like it just depends on the $m$-th homogeneous component.

I must be doing the substitution wrong. After setting $\tau = t/r$, don’t you pick up a factor of $r^{-2s}$ in front, not $r^s$? The text also seems to reverse the bounds of integration between line 2 and 3.

This is up to opinion, but I do think that some important mathematical ideas can resist formalization, or at least are not yet mature enough to formalize. Instead they manifest as general principles which pop up in so many diverse contexts, that any attempt to formalize them greatly constrains their applications and can even limit their clarity. Frequently these are waiting for the proper general setting to state them in. This does not prevent experts from fruitfully discussing and applying them.

…no branch point, so this undercounts by 1. It is difficult for me to evaluate this in modern language, but my superficial guess is that it is more or less the same concept as your argument above— just that looking dually at holomorphic forms rather than meromorphic functions clarifies the role of the branch points/values in the count.

@AlexandreEremenko Thank you. He does note that the argument fails for g = 1 and gives a separate argument: as far as I can tell, he talks about inducing a holomorphic function on the universal cover of the surface by integrating a holomorphic form along paths. This holomorphic form has $g$ degrees of freedom, which permit us to fix the integrals over $g$ out of $2g$ generators of $\pi_1$, and the induced function has another by adding a constant, which controls one of the branch values of the function. $2g$ generators + $2g-2$ branch points - $(g+1)$ parameters = $3g-3$. When $g=1$ there is…

@AlexandreEremenko I apologize if it is poor form to resurrect such an old answer, but I am struggling to understand the distinction that prevents this reasoning from holding when $g=1$. I have skimmed Riemann’s paper and am working to interpret it in my knowledge of the modern language. It seems that there is some subtlety in the “arbitrary assignment” of branch values, and the question of whether such assignment uniquely determines a surface and a map from it. If possible could you please elaborate on this point?