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Where can I learn about the reduction of the Jacobians of modular curves such as X_0(N) and X_1(N) at primes p dividing N?


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Springing to mind is the Conrad-Edixhoven-Stein paper "$J_1(p)$ has connected fibers". Do you know about bad reduction of the curves? –  stankewicz Oct 21 '11 at 0:22
Edixhoven's Trieste notes (math.leidenuniv.nl/~edix/public_html_rennes/cours/trieste.html) might not contain what you are looking for but they are an excellent succinct introduction to the theory. –  Chandan Singh Dalawat Oct 21 '11 at 4:46
@Chandan Thanks for the link to those notes. They are really good! –  Keenan Kidwell Oct 22 '11 at 1:36
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1 Answer

up vote 16 down vote accepted

The standard places that one learns this are (or at least, used to be): Mazur's Eisenstein ideal paper (which treats the case of $X_0(N)$ for $N$ prime in great detail), Ribet's papers (his Herbrand criterion paper, his Warsaw ICM talk, his Inventiones 100 paper, and several others as well), Gross's Tameness criterion paper, and the papers of Mazur--Wiles. Most of these references (Mazur--Wiles is one exception) tend to focus primarily or even exclusively on the case when $p$ exactly divides $N$. One reason for this is that, in the case when $p$ exactly divides $N$, the Jacobian is semi-abelian when reduced mod $p$ (i.e. an extension of an abelian variety by a torus) (at least after making a base-change, in the $\Gamma_1$ case). The situation is worse when $p^2$ divides $N$.

There is another way to think about this question, which is not typically how people learned it when I was a student, but might be a better way to proceed nowadays. Namely, $J_0(N)$ or $J_1(N)$ is (up to isogeny) a product of abelian varieties $A_f$ attached to Hecke eigenforms $A_f$. Now the reduction of $A_f$ is (more-or-less) determined by the amount of ramification at $p$ in the $\ell$-adic Tate module of $A_f$, which is to say, in the ramification at $p$ of the $\ell$-adic Galois representation $\rho_f$ attached to $f$. (Here $\ell$ can be any prime distinct from $p$.)

Now the restriction to a decomposition group at $p$ of $\rho_f$, and so, in particular, the ramification at $p$ in $\rho_f$, is determined by the the local factor at $p$ of the automorphic representation generated by $f$. (This is the celebrated local-global compatibility theorem of Langlands--Deligne--Carayol.) This local factor encodes information such as the power of $p$ dividing the level of $f$ and the eigenvalue of $U_p$ acting on $f$ (but in general it contains more information than just these facts).

A typical concrete consequence of this analysis is that if $p$ is prime to $N$, and if we write $J_1(N;p)$ to denote the Jacobian of the modular curve of level $\Gamma_1(N)\cap \Gamma_0(p)$, then $J_1(Np)/J_1(N;p)$ has good reduction over $\mathbb Q_p(\mu_p)$.

Of course, the two methods aren't really different --- the arguments with arithmetic geometry and moduli of elliptic curves that appear in the first stream of literature mentioned are also used to make the deductions about local-global compatibility in Carayol's paper. But the automorphic viewpoint of Langlands--Deligne--Carayol is more powerful, and also more efficient, in some sense: it collects all the geometry in one argument, and then phrases the conclusions in the very efficient "local-global compatibility" language. Unfortunately, the literature in this stream is a little more austere than in the first stream, and the first stream also benefits from the fantastic expositional skills of Ribet and Gross especially.

In conclusion, I might begin with Ribet's Inventiones 100 paper and Gross's Tameness criterion paper, and after this, try to understand the statements of Carayol's local-global compatibility results (and see how to recover the results about Jacobians in Ribet and Gross's papers from Carayol's more general statements).

One more thing: Serre and Tate's paper On the good reduction of abelian varieties is a terrific read, and is good for learning how the Galois action on its Tate modules influences the reduction type of the abelian variety.

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Thanks for the useful answer, Matt. –  Nicolás Oct 21 '11 at 13:56
I don't know much about this, but I saw Jared Weinstein give a great talk on the case where $N$ is divisible by a high power of $p$. His preprint is arxiv.org/abs/1010.4241 ; you might want to look there if you need to get into the details of this case. –  David Speyer Oct 21 '11 at 18:42
Dear David, Yes, in fact Jared's goal is to understand Carayol's local-global compatibility in explicit geometric terms. Regards, Matt –  Emerton Oct 22 '11 at 4:48
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