Why is there a weight 2 modular form congruent to any modular form

Question

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.

Answer 1

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.

Answer 2

If $N\geq 1$ is an integer not divisible by $p$, one can see that any system of Hecke eigenvalues $(a_\ell)$ arising from $S_k(\Gamma_1(N))$ is congruent mod $p$ to a system $(b_\ell)$ arising from $S_2(\Gamma_1(Np^n))$, for some $n$, using an interplay between a theorem of Serre (describing a purely mod $p$ Jacquet-Langlands correspondence), and the more classical, characteristic zero J-L between ${\rm GL}_2$ and the multiplicative group $G$ of the $\mathbf{Q}$-quaternion algebra ramified at $p$ and infinity.

I know that what I describe here is perhaps not the right way of proving the result you are asking, but it seems to me worth to mention.

In his '87 letter to Tate Serre proves:

${\rm Theorem:}$ Systems of mod $p$ Hecke eigenvalues arising from $M_k(\Gamma_1(N))$ are the same as those arising from locally constant function $f:G(A)\rightarrow\overline{F}_p$ that are left invariant under $G(\mathbf{Q})$ and right invariant under a certain open subgroup $K_N$.

Here $G(A)$ is the adelic group associated to $G$. Notice that the functions considered on the quaternion side are independent of the archimedean variable. Moreover, the double coset $G(\mathbf{Q})\backslash G(A)/K_N$ is finite and any mod $p$ system of eigenvalues arising from it can be lifted to characteristic zero.

Therefore applying the theorem and then lifting, we see that for any (char. zero) eigensystem $A=(a_\ell)$ arising from $M_k(\Gamma_1(N))$ there is a (char. zero) eigensystem $B=(b_\ell)$ arising from the space of locally constant functions $f:G(A)\rightarrow\mathbf{C}$ such that $A\equiv B$ mod $P$, where $P$ is a fixed prime of $\overline{\mathbf{Z}}$ lying over $p$.

Assuming that the automorphic form $\Pi_B$ on $G$ associated to $B$ is infinite dimensional, by the J-L correspondence we have that there is a cuspidal automorphic form $\Pi'_B$ on ${\rm GL}_2$ associated to the same eigensystem $B$. The type of $\Pi'_B$ at any finite place other than $p$ is the same as that of $\Pi_B$, while at infinity $\Pi'_B$ is the discrete series of lowest weight $2$. This basically says that there is a cusp form in $S_2(Np^n)$ whose associated system of eigenvalues is $B=(b_\ell)$.

We are only left with deciding when $\Pi_B$ is infinite dimensional, or can be chosen as such. This happens only for systems of eigenvalues of the form $B=(\chi(\ell)(1+\ell))_{\ell\nmid pN}$, where $\chi:\mathbf{Z}/p\rightarrow\mathbf{C}^*$ is any character (in order to show this one has to consider the particular shape of $K_N$, which I did not even define..). The reduction mod $P$ of such eigensystems are all of the form $(\ell^k+\ell^{k+1})_{\ell\nmid pN}$.

Concluding: Let $A=(a_\ell)$ be a sytstem of char. zero eigenvalues arising from $M_k(\Gamma_1(N))$, with $p\nmid N$. Assume that the mod $P$ reduction of $A$ is not of the form $(\ell^k+\ell^{k+1})_{\ell\nmid pN}$. Then, there exists a cusp form in $S_2(\Gamma_1(Np^n))$ such that its associated system of eigenvalues $B$ is congruent to $A$ mod $P$.