# Good introductory references on moduli (stacks), for arithmetic objects

I've studied some fundation of algebraic geometry, such as Hartshorne's "Algebraic Geometry", Liu's "Algebraic Geometry and Arithmetic Curves", Silverman's "The Arithmetic of Elliptic Curves", and some chapters of Mumford's "Abelian Varieties". I would like to learn more advanced arithmetic, and I began reading Faltings's chapter, which shows the Faltings's theorem (Mordell conjecture), in Cornell, Silverman's "Arithmetic Geometry". But this chapter needs moduli stacks for curves and abelian varieties.

I've heard that Harris, Morrison's "Moduli of Curves" is a good reference. But glancing through some sections, I think this book seems to concern only curves over $\mathbb{C}$. So the first question is: is this book enough for my interesting? That is, do arithmetic geometers use moduli of curves only over $\mathbb{C}$? If not, please suggest to me good references.

And I've heard that Mumford's GIT is good, for moduli of abelian varieties. Although these books discusses only moduli schemes, I would like to learn moduli stacks. So the second question is: please suggest to me some references for moduli stacks. And is it a good way to learn moduli stacks, not learning moduli schemes?

Finally, I'm completely beginner of this field, but I intuitively think: for a functor $\mathscr{F}$ that takes a scheme $S$ to the set of isomorphism classes of some objects (e.g., smooth projective curves of given genus, or principally polarized abelian varieties of given dimension) over $S$, the fine moduli is the representable scheme of $\mathscr{F}$ (in general, there does not exist), and the coarse moduli is the "almost" representable scheme of $\mathscr{F}$, and the moduli stack is the representable object of $\mathscr{F}$, in some extended category. Is this idea wrong?

Thank you very much!

• For a first introduction I would very much recommend Picard groups of moduli problems by David Mumford (Arithmetical Algebraic Geometry pp. 33-81, Harper & Row, New York). Though the word "stack" is not pronounced, it is a beautiful study of the moduli stack of elliptic curves, which shows how one can work with a stack and why it is a natural thing to do. – abx Aug 28 '18 at 5:04
• I found Martin Olsson's book "Algebraic Spaces and Stacks" to be quite good as well. – Will Chen Aug 28 '18 at 5:58