Hi,


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?


Thanks!



References for bad reduction of Jacobians of modular curves?
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.


