# 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. Aug 28 '18 at 5:58

If you want to learn about stacks, I can recommend 'Fundamental Algebraic Geometry: Grothendieck's FGA Explained'. Vistoli's exposition of the basic theory of stacks is hard to beat, I think. Moreover, the chapter about Picard schemes is also good if you want to learn when a functor is representable and what you might have to do to make it representable. In any case, the theory of Picard schemes is indispensable in arithmetic geometry, e.g. it is a way to obtain the group structure on an elliptic curve (see the book of Katz, Mazur: Arithmetic moduli of elliptic curves) or to study duality of abelian varieties.

I am not sure if there exists an ideal book if you want to learn about the moduli stack of elliptic curves. The classic source is Deligne-Rapoport 'LES SCHEMAS DE MODULES DE COURBES ELLIPTIQUE', which you should look into, but the proofs are often very brief. The book by Katz and Mazur has good parts, but for some reason they decided to avoid the language of stacks. The book by Olsson on stacks has some parts about elliptic curves as well. At some point, I wrote a note together with Viktoriya Ozornova that contains a really detailed proof that the moduli stack of elliptic curves is really an fpqc stack and that Weierstraß equations exist http://www.staff.science.uu.nl/~meier007/Mell.pdf (but it is certainly not meant to provide a flair of the subject).

• In the proof of lemma 2.5.(3) of your note, using Stack Project Tag0CC1, you conclude that $\operatorname{Spec}R \to \operatorname{Spec}R_0$ has the dense image. But this proposition assumes that $R$ is an integral domain. Why do we use this for general $R$?
– k.j.
Jun 10 '20 at 2:48
• @k.j. The citation is indeed wrong, but the statement that an injective morphism of rings induces a dominant morphism on their spectra is still true. See for example math.stackexchange.com/questions/389036/… for a proof. Jun 10 '20 at 6:56

A very nice introduction to moduli (stack) of elliptic curves is R. Hain - Lectures on Moduli Space of Elliptic Curves (you find it on arXiv):

1. it is completed by 100 and more exercises;
2. a brief appendix is devoted to stacks (I suggest to integrate it with Vistoli's lecture notes on stacks);
3. it doesn't introduce the elliptic curves, therefore you must know it (but this is not the case of the OP).