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As noted earlier, I found reading Weil's book "Basic Number Theory" to be a harrowing experience, and I find his writing to be intrinsically hard to understand, though it is perfectly rigorous and cle …

First case: Complex numbers. Over $\mathbb C$ the structure as an abstract group is $\mathbb S^1 \oplus \mathbb S^1$ where $\mathbb S^1$ is the circle, i.e., $\mathbb R/\mathbb Z$. This follows as Rob …

I saw that there are many "applications" questions in Mathoverflow; so hopefully this is an appropriate question. I was rather surprised that there were only five questions at Mathoverflow so far with …

Let $p$ be a prime. Suppose you have an Abelian scheme $A$ over $Spec\ \mathbb{Z}_p$. How do you prove that if $q$ is another prime, then the $q$-torsion of $A$ injects into the torsion of $A_p$, unde …

I find the following description of Artin motives in Wikipedia. Since these seem to be quite related to number theory, I am interested to learn more in that context. I request the experts available in …

Background: Once an analytic number theorist remarked to me that all attempts to prove the Riemann hypothesis using number theoretic methods have failed. Since then that remark stuck in my mind. The …

This is a speculation and perhaps naive. The theorem of Siegel that There exist only finitely many integral points on a curve of genus $\geq 1$ over a number ring $\mathcal O_{K, S}$ where $S$ is …

Compared to Fermat's two squares theorem, or Fermat's four squares theorem, Fermat's Little theorem is indeed Little. Not to mention the hard-to-prove Fermat Last Theorem, which goes under FLT; so th …

It is apparent that the abc conjecture is deeply related to Arakelov theory. In one direction, it is shown in S. Lang, "Introduction to Arakelov Theory", that a certain height inequality in Arakelov t …

The article I have checked for Baker's theorem is Waldschmidt's. But the article and the citations therein are from the time of '88. Question: What is the the strongest known lower bound for Baker …

In number theory, base change of a scheme or a variety is with respect to the underlying ring or field, is viewing the same scheme/variety over an extended ring or field, but with the "same" set of eq …

I do not know about algebraic number approximations; but the most canonical approximation to real numbers by rational numbers is the continued fraction expansion. They and their convergents (of first …

This answer is tangential in the sense that it is speaking of the existence of solutions rather than counting them all. But I rather suspect that you would find this interesting. Suppose you have a q …

When I read it for the first time, I found the whole slog towards proving the Prime Number Theorem and the final success to be magnificent. So I am curious about more general results. We talk of comp …

Try Serre, Topics in Galois Theory.