So, arithmetic geometry studies objects like varities over any field not necessarly algebraiclly closed and it seems that diophantine geometry is closely related and I'm concerned about the relation of the two and their differences regarding the object of study, the questions they ask and the methods by which they try to answer such questions.

Algebraic number theoey seems to be closely related to arithmetic geometry and I would like to know how things fit together.

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    $\begingroup$ A very open-ended question. I’d like to see it more narrowly set out. $\endgroup$ – Lubin Feb 14 '18 at 1:12

Here's a rough attempt at an answer.

Arithmetic geometry is what happens when you do algebraic geometry over bases that have interesting number-theoretical properties, usually $\operatorname{Spec}\Bbb Z$ or finite fields (maybe $p$-adics too, depending on how much you consider that it's own thing).

Diophantine geometry: a synonym for arithmetic geometry, especially in the case that you have an affine scheme of finite type over $\operatorname{Spec}\Bbb Z$.

Alegbraic number theory is more or less anything that can be generalized out of class field theory: subjects that arise out of the study of finite abelian field extensions of number fields.

It happens that algebraic number and arithmetic geometry converge a fair bit, especially in that higher class field theory can see some rephrasing in terms of algebraic K-theory, which is pretty deeply attached to some difficult algebraic geometry. I'm not an expert in the forefront of research in either area, but I do know some fellow graduate students focusing in those areas, and it seems that they often switch handily between an algebro-geometric and number-theoretic perspective in their work.

It should be said that a lot of things which can be approached in an algebraic manner in mathematics can get subsumed under the heading of algebraic geometry - algebraic geometers have worked hard on making their subject able to tackle lots and lots of problems since the 1950s. Arithmetic geometry is more or less the product of applying this philosophy of having algebraic geometry do all the things to algebraic number theory.

I've made this post community wiki so that a better answer can be generated, because I know there are more satisfying explanations out there (and I encourage others to add to this).


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