# Arithmetic geometry from a bird's-eye view

Is ist true that Arithmetic Geometry can roughly be separated into two areas: 1) Showing that motivic $L$-functions are automorphic. 2) Calculating special values of these $L$-functions.

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It seems to me that many famous results in Arithmetic Geometry are about bounding the number of rational points on varieties. Which category would this fall into? –  David Speyer Sep 21 '10 at 13:07
If one speaks of Abelian varieties, it would fall into category 2) (BSD conjecture). I don't know if the ETNC would help in case of arbitrary motives. –  Timo Keller Sep 21 '10 at 13:41
No. 1) and 2) are very laudable pursuits but are just a tiny part of what I'd consider arithmetic geometry. I'd say that what I've been doing in the last 25 years is mostly arithmetic geometry and I've never done either of those things. –  Felipe Voloch Sep 21 '10 at 14:03
Where does the arithmetic theory of linear algebraic groups (e.g.., Hasse principle in simply connected case, Galois cohomological classification of forms, etc.) fit into your dichotomy? Or $p$-adic Hodge theory? Or the Mordell conjecture? Or the semistable reduction theorem for curves? –  BCnrd Sep 21 '10 at 14:27

No, I think your suggested dichotomy misses too many vitally important results in arithmetic geometry. In particular, there are purely Diophantine questions (and answers) which are not informed by any automorphic/motivic considerations whatsoever. As a point of philosophy, you might want to speculate that some of these results will eventually have automorphic interpretations, but in many cases there is not even presently a conjectural relation.

To give a specific example, I will go out on a limb and claim that the single greatest theorem in arithmetic geometry is Faltings' proof of the Mordell-Lang Conjecture (informally, I like to think of this as the Faltings-Vojta-Faltings theorem): if $A$ is an abelian variety over a number field $k$, and $\Gamma \subset A(\overline{k})$ is a subgroup such that $\dim_{\mathbb{Q}} \Gamma \otimes \mathbb{Q} < \infty$, Then for any closed subvariety $X \subset A$, there exists $n \in \mathbb{Z}^+$, $\gamma_1,\ldots,\gamma_n \in \Gamma$ and abelian subvarieties $B_1,\ldots,B_n$ of $A$ such that

$\Gamma \cap X(\mathbb{C}) = \bigcup_{i=1}^n \gamma_i + (B_i(\mathbb{C}) \cap \Gamma)$.

In particular, this reduces Faltings' earlier finiteness theorem (finiteness of $k$-rational points on a curve $X$ of genus at least $2$) to the Mordell-Weil theorem and recovers the Manin-Mumford conjecture (that a curve of genus at least $2$ embedded in its Jacobian contains only finitely many torsion points).

Some good articles on the subject include:

(The last one gives a signficantly more general result!)

If anyone can relate this seminal result to motivic/automorphic anything, I would be very interested to know.

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No, because there is the area called Diophantine geometry, and the formulation assumes it can be absorbed into "non-abelian class field theory" and "algebraic K-theory". We don't know that it can't, if you take the scare quotes broadly enough. But the theory that it can seems to me like one of those mathematical "mergers and acquisitions" deals that appeats good and fashionable on paper, for a while. If however you look at those three just for 30 seconds, you may see the old algebra-analysis-geometry trio staring back at you (automorphic theory for the analysis, the geometric approach to equations being what Diophantine geometry has as its whole rationale). And then you might decide you have seen this one before.

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