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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 ...

(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 $3X^3 + 4Y^3 + 5Z^3 = 0$. ...

The analogy between $O_K$ ($K$ a number field) and affine curves over a field has been very fruitful. It also knows many variations: the field over which the curve is defined may have positive or zero ...

I am always fascinated when a quadratic form (or a quadric) arises naturally. I have some elementary examples, but most of all, I want to learn more examples. I hope this question isn't considered too ...