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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 …

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 …

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 …

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 …

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 …

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 …

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 …

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 …

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 …

See Chandrasekharan, Analyic Number Theory, for the proof by S. S. Pillai. It is quite easy.

Your question about one book for number theory is like a non-mathematician asking about one book for all mathematics. It is simply not possible. It is a growing subject in various directions. The best …

The Anterp conference volumes, "Modular functions in one Variable - I, II, III, .... " might contain what you want. I am not sure though, as I am unable to verify it by looking into all the volumes.

Look at Darmon's article on "Arithmetic of Curves".

Use the PARI/GP routine ellglobalred. See here for a list of elliptic curve routines for PARI. I copy the relevant part here, from that page: ellglobalred(E) calculates the arithmetic conductor, the …