I got my copy of Computational Aspects of Modular Forms and Galois Representations in the mail yesterday. The goal of the book is "How one can compute in polynomial time the value of Ramanujan's tau at a prime", well, or any other modular form of level 1. It's all very thrilling!


The following fact is essential: for any modular form $f$ of level 1, and any prime $l$, the mod $l$ reduction of the semisimplification of the galois representation attached to $f$ by Deligne is a 2-dimensional subrepresentation of the galois representation of the $l$-torsion of the jacobian of the modular curve of level $l$.


Why?


From what I understand, this is somewhat equivalent (after Shimura and Deligne) to there being a modular form of weight 2 and level $l$ that is congruent to $f$ mod $l$, or something similar. Is this the right statement? Why is it true then?


Searching far and wide for an introduction to this topic yields very little.



Why is there a weight 2 modular form congruent to any modular form
By "level $\ell$" I assume you mean "level $\Gamma\_1(\ell)$".


Here's a proof. By the Eichler-Shimura theorem, the system of eigenvalues associated to the modular form shows up in $H^1(SL(2,\mathbf{Z}),Symm^{k-2}(\mathbf{C}))$. Hence (by some easy commutative algebra) the mod $\ell$ reduction of the system of eigenvalues shows up in $H^1(SL(2,\mathbf{Z}),Symm^{k-2}(\mathbf{F}\_\ell))$ and hence, by a standard diagram chase, in $H^1(SL(2,\mathbf{Z}),M)$ for $M$ an irreducible module for $GL(2,\mathbf{F}\_\ell)$ (EDIT: here $M$ is a finite-dimensional vector space over $\mathbf{F}\_\ell$, so it's just a twist of $Symm^n$ for some small $n$). But any such $M$ is a subquotient of $I:=Ind\_{(\* \*;0 1)}^{GL(2,\mathbf{F}\_\ell)}(1)$ so the system of eigenvalues shows up in $H^1(SL(2,\mathbf{Z}),I)$ and hence, by Shapiro, in $H^1(\Gamma\_1(\ell),1)$. (EDIT: here $1$ means the trivial 1-d vector space over $\mathbf{F}\_\ell$: one now deduces that the system of eigenvalues lifts to a system of evals showing up in $H^1(\Gamma\_1(\ell),\mathbf{C})$).


Now using Eichler-Shimura again, this time at level $\ell$, shows that there's a weight 2 level $\Gamma\_1(\ell)$ modular form giving rise to the same mod $\ell$ system of Hecke eigenvalues. This last statement is a little disingenuous because Eichler-Shimura only tells you about parabolic cohomology which isn't quite the same as group cohomology. But the extra stuff is all associated to reducible Galois representations so can be dealt with by hand using Eisenstein series.


You'll find these sorts of arguments in papers of Ribet from around 1987-1990. Another great place to look is papers of Ash and Stevens from slightly earlier -- I learnt the argument below from an Ash-Stevens paper.


