# Whats the difference between arithmetic geometry and algebraic geometry?

both seem to be about geometry. why the distinction? I mean which preceded the other? Why is algebraic geometry more popular?

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This question is too broad. There are plenty of ways to get a basic understanding using Google. – Qiaochu Yuan Aug 18 '11 at 0:39
Algebraic geometry has many more applications. – André Nicolas Aug 18 '11 at 1:09
Is this second comment for real? I hope it is just a joke. – Matt Aug 18 '11 at 1:11
I never heard the term "arithmetic geometry" until maybe 20 years ago, roughly the time of Wiles' work on Fermat. The term "algebraic geometry" goes back quite a bit farther. – Gerry Myerson Aug 18 '11 at 1:15

I don't know when the subject name "arithmetic geometry" (or sometimes "arithmetic algebraic geometry") was first coined, but it refers (more or less) to the study of Diophantine equations via methods of algebraic geometry.

To speak of "which preceded the other" is perhaps not so sensible, since, while the name arithmetic geometry is relatively modern, on the one hand, a geometric approach to solving certain Diophantine equations goes back perhaps to Fermat, or even Diophantus in some instances, while on the other hand algebraic geometry (again a fairly modern name) is a rich subject with its own deep legacy and heritage.

From a modern (rather than historical) perspective, algebraic geometry is the broader field, while arithmetic geometry is a part of algebraic geometry (the part that intersects with number theory).

Algebraic geometry as a term covers many more different areas of mathematics than arithmetic geometry. (As I already indicated, the latter is a subset of the former.) Given this, it is not surprising that there are more algebraic geometers than arithmetic geometers. I've never thought this was a question of popularity, though.

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So are "arithmetic geometry" and "arithmetic algebraic geometry" interchangeable terms? Is there ever a distinction? I was recently told - to my utter chagrin - that the two subjects were distinct and that the latter was more complicated. – PrimeRibeyeDeal Sep 10 '13 at 21:08
@PrimeRibeyeDeal: Dear PRD, I don't know how I would distinguish one from the other. I guess I would have to ask which mathemticians they think of working in each of the (allegedly) two subjects. E.g. is Faltings an arithmetic geometer or arithmetic algebraic geometer? What about Scholze? What about Kisin? I think you'll be hard pressed to find any rational reason for choosing one description over another for any of them. Regards, – Matt E Sep 11 '13 at 3:35
I recently heard a possible distinction: arithmetic algebraic geometry tends to deal the arithmetic of algebraic varieties (reduction mod p, etc.), especially abelian varieties, in which case an integral model may not be provided or is not the main focus, whereas straight up arithmetic geometry focuses primarily on those integral models, the (especially flat) schemes over $Spec\mathbb{Z}$. However it's possible that Diophantine geometry is actually the proper term for the first description I gave. Would anyone care to comment? – PrimeRibeyeDeal Dec 12 '15 at 21:42