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Let $X$ be a complex, projective algebraic variety and assume that $X$ has a model $X_0$ over $\mathbb Z$ i.e. $X\cong X_0\times_{\operatorname{Spec }\mathbb Z}\operatorname{Spec }\mathbb C$.

Let's take a look at the following quote from the book "Mumford, Oda - Algebraic Geometry II" (In what follows we can put $\mathbb Z=R$): Check this picture.

So, the aim of the "Kroneckerian geometry" is to find relations between the (classical) geometry of the complex variety $X$ and the arithmetic (for example Diophantine properties) of the $\mathbb Z$-model $X_0$.

I'd like to collect a partial list of the most important results in this direction. In more practical terms I'm asking you to write the best (or your favourite) examples of "Kroneckerian geometry in action".

Many thanks in advance

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  • $\begingroup$ Your first statement needs to be more precise. For example, usually one assumes that $X_0$ is projective over $\mathrm{Spec}(\mathbb{Z})$. Moreover, such an $X_0$ always exists (but is not unique) so additional assumptions may be put in to avoid "junk" in the closed fibres of $X_0$ over $\mathbb{Z}$. $\endgroup$
    – Kapil
    Commented Dec 16, 2019 at 20:32
  • $\begingroup$ Yes, I suppose we want $X_0$ projective. You are right. $\endgroup$
    – Paolo Dolce
    Commented Dec 16, 2019 at 21:16

1 Answer 1

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For curves, the $\mathbb Z$ points (equivalently the $\mathbb Q$ points) are finite if the genus (in the topological sense) of the complex riemann surface is at least $2$. This is Mordell's conjecture, aka Falting's theorem.

In the genus $0$ case, the rational points are either empty or infinitely many.

In the genus $1$ case the rational points form a finitely generated abelian group with possibly $0$ rank. This is Mordell-Weil.

This also works for affine curves (and with any number ring instead of the integers) - the rational points are finite if the Euler characteristic is negative (aka hyperbolic geometry).

I think you also want your model to be flat to get good behavior (this is automatic in dimension one).

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