Questions tagged [arithmetic-geometry]

Despite appearances, this is not a question on computational aspects of number theory. The background is as follows. I once asked a number theorist about what he considered to be the most important unsolved problems in arithmetic geometry. He told…

The arithmetic–geometric mean of 2 values $a_0$,$b_0$, is the value to which the arithmetic and geometric values converge, being $$a_n=\frac{a_{n-1}+b_{n-1}}{2} \text{ and } b_n=\sqrt{a_{n-1} .b_{n-1}}$$ with $$\lim_{n \to \infty} a_n = \lim_{n \to…

This is an exercise (1.15) of Dino Lorenzini' s An Invitation to Arithmetic Geometry. Let $F$ be a field of characteristic $2$. Let $k:=F(t)$, the field of rational functions in the variable $t$. Let $f(x)=x^6+tx^5+t^2x^3+t\in k[x]$. We want to show…

By arithmetic surface we mean a projective, regular and flat scheme of dimension 1 over $Spec(O_K)$ for some algebraic number field $K$. So we can view such a scheme $X$ as a closed subscheme of $Proj(O_K[T_0,...,T_n])$ or equivalently…