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The search turned up a 1981 paper by John S.Lew (in the Unsolved problems section) Polynomials in Two Variables Taking Distinct Integer Values at Lattice-Points which discusses related problems, and …

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 $\ …

Robert Langlands is famous in number theory for making famous and deep conjectures about very abstract things called automorphic forms, somewhere in the 60s. There's a very interesting article by Lan …

I'm sure you omitted this just because it's too classic: big part of group theory was invented to prove that most algebraic numbers cannot be constructed by radical extensions. It's still the best, …

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 …

There are several related questions here, the latter being especially interesting. We know the classical Eisenstein series. What are the Eisenstein series on a group G and why they are interesting? …

There's a very famous group, the largest sporadic simple finite group, sometimes called a monster whose size is quoted below. What's the explanation that the primes appearing in it, #{Monster} = 2 …

I thought that the order of the Tate-Shafarevich group should always be a square (it's also supposed to be finite, but for the purposes of this question let's assume we know this) but I don't seem to …

Sure, here's a overview. Suppose you have a ring R over a field k, then, by the magic of algebraic geometry, you can think about it in a geometric way. You do this by defining points as epimorphisms …

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 …

It's well-known that every natural number can be written as a sum of 4 squares of integers. Has there been any recent progress about the similar problem for the cubes, 4-th powers and so on? I believ …

I've been reading the wonderful slides by Terry Tao and thought about this question. Primes appear to be quite random, and the formal statement should be that there are some characteristics of primes …

My goal was to learn about l-adic representations on some example — I'm a newbie in these topics. Thus take pt = Spec F_q, G=\pi_1(pt) and consider lisse schemes over pt. My understanding is that suc …

Trying to understand answer to this question. What is the (Beilinson) higher regulator of a number field?

Are supersingular primes and supersingular elliptic curves related? (this was essentially a subquestion in my earlier question, but still looks sufficiently different to me to deserve a separate post …