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I'm not aware of such a theorem, but you should take a look into absolute zeta-functions, which hope to prove a lot about zeta-functions — perhaps up to Riemann's Hypothesis. Since I don't know much …

It appears that you want the "rough, introductory answers", so feel fine to ask more detailed questions if anything is unclear/potentially wrong. Also, I only have very general knowledge about enumera …

You can also take a look at the question A learning roadmap for algebraic geometry.

In standard topological terms, the exact sequence that relates homotopy groups of the base $B$, fiber $F$ and total space $E$ of topological fibration gives $$\pi_1(F) \to \pi_1(E) \to \pi_1(B) \to \ …

You can do a reverse construction: start with a sphere without 4 points; now add two points over each one in such a way that every time you go around one hole the two points get interchanged. The sam …

Source This question came up in the discussion between Kevin Buzzard and Minhyong Kim in the comments to Smooth proper scheme over Z. It was 2 weeks ago, so I took the liberty of posting it as commun …

Preliminaries First part of your question doesn't use the bialgebra structure. That is, you have a space of functions on countable many points which I'll denote $A = \mathbb C_1\times \mathbb C_2\tim …

I don't have much knowledge in either knots or Hopf bialgebras, but I think one can answer part of your question (the one formerly asked as a title: "What do negative knots look like?") anyway. Abou …

How do you define blow-up? It should be straightforward to show that an explicit construction has relative dimension 0 over $X$. Update: In the comments I suggest to take the formal two-dimensional …

Comments by Anweshi The essential point is what Emerton mentioned, ie the analogy with Minkowski's theorem on number fields with ramification. The basic principle is that "arithmetic is geometry". Nu …

Motivation I learned about this question from a wonderful article Rational points on curves by Henri Darmon. He gives a list of statements (some are theorems, some conjectures) of the form the set $\ …

It is not sufficient that the subscheme of poles and zeroes is defined together over $K$, as the example of the function $(z+i)/(z-i):\mathbb P^1 \to \mathbb P^1$, defined only over $\mathbb Q(i)$, il …

Yes, this is a big area of research. I'll add some references to the ones Dmitri provides. Here are references from a question about Moment map for toric actions: Riemann-Roch for toric orbifolds b …

I had this simple question when formulating the Todd class question. Does there exist an example of proper morphism $f:X\to Y$ together with nontrivial homology class $t\in H^*(X)$ such that for a …

Suppose for some reason one would be expecting a formula of the kind $$\mathop{\text{ch}}(f_!\mathcal F)\ =\ f_*(\mathop{\text{ch}}(\mathcal F)\cdot t_f)$$ valid in $H^*(Y)$ where $f:X\to Y$ is a …