All Questions

As the title suggests I am interested in CRT applications. Wikipedia article on CRT lists some of the well known applications (e.g. used in the RSA algorithm, used to construct an elegant Gödel ...

The question is about characterising the sets $S(K)$ of primes which split completely in a given galoisian extension $K|\mathbb{Q}$. Do recent results such as Serre's modularity conjecture (as proved ...

The projective curve $3x^3+4y^3+5z^3=0$ is often cited as an example (given by Selmer) of a failure of the Hasse Principle: the equation has solutions in any completion of the rationals $\mathbb Q$, ...

(This is inspired by Algebraic geometry examples.) I want to collect here (counter)examples in arithmetic geometry. Curves violating the Hasse principle: The Selmer curve $3X^3 + 4Y^3 + 5Z^3 = 0$. ...

Does anyone have examples of when an object is positive, then it has (or does not have) a square root? Or more generally, can be written as a sum of squares? Example. A positive integer does not ...

Recently the elliptic curve $E:y^2+y=x^3-x^2$ of conductor $11$ (which appears in my answer) became my favourite elliptic over $\bf Q$ because the associated modular form $$ F=q\prod_{n>0}(1-q^n)^2(...