## Even Galois Representations mod p

Suppose that $\overline{\rho}: G_{\mathbf{Q}}: \rightarrow \mathrm{GL}_2(\overline{\mathbf{F}}_p)$ is a continuous  irreducible Galois representation. What does the Langlands program say about such $\overline{\rho}$? When $\overline{\rho}$ is odd, the situation is quite satisfactory, the answer being given by Serre’s conjecture. For example, having fixed a Serre weight $k \ge 2$ and a Serre level $N$, one knows that there will only be finitely many such representations and they will all come from classical modular forms for $\mathrm{GL}(2)$.

When $\overline{\rho}$ is even, however, there is an equally good (if conjectural) description of such representations. First, the dihedral representations are well understood by class field theory, so let us assume we are not in this case. Then, replacing $\overline{\rho}$ by the adjoint representation $\mathrm{ad}^{0}\overline{\rho}$ and restricting to some (any) imaginary quadratic field, one obtains an irreducible (conjugate) self-dual representation, which, by the generalization of Serre’s conjecture, should come from an automorphic representation for $U(3)$. It follows that, as in the odd case, there will (conjecturally) only be finitely many such $\overline{\rho}$ for a fixed pair $(N,k)$. However, things are even better in the even case. Namely, if one fixes $(N,k)$ but allows $p$ to vary, then there will still only be finitely many even representations, in contrast to the odd case where (for $(N,k) = (1,12)$ for example) such representations occur for infinitely many $p$. The reason is that all such representations will have to arise from a fixed finite dimensional space of automorphic forms determined by $N$ and $k$, and thus (by the pigeonhole principle) there will exist an automorphic $\Pi$ for $U(3)$ whose $\mod p$ representation extends to an even representation of $\mathbf{Q}$ for infinitely many $p$. By multiplicity one, it would follow that $\Pi \simeq \Pi^c \simeq \Pi^{\vee}$ and hence $\Pi$ itself must come from  the adjoint representation of a form from $\mathrm{GL}(2)$ over $\mathbf{Q}$, which would imply (since we are in regular weight) that the representations are odd. Note that it is important in the definition of Serre weight here that $k \ge 2$; if one allows $k = 1$ then there exist representations in characteristic zero which give rise to mod $p$ representations for all $p$.

Here’s a specific example in which one can prove finiteness. Suppose that we consider representations with $k=2$ and $N=1$. Then there are no such even $\overline{\rho}$ for a stupid reason, because the determinant will be cyclotomic (Tate deals with the case $p=2$.) Now consider the case when $k=2$ and $N=4$.  In the even case, the determinant must be the cyclotomic character times the unique (odd) character of conductor $4$. Let’s prove that there are no such representations. Tate like arguments reduce to the case when the representation has image containing $\mathrm{SL}_2(\mathbf{F}_p)$ and $p \ge 7$. Now take the auxiliary imaginary quadratic field to be $\mathbf{Q}(\sqrt{-1})$. The corresponding adjoint representation now is unramified outside primes above $p$ (the quadratic extension eliminating the ramification at $2$) and is Fontaine-Laffaile with weights $[-1,0,1]$ at primes dividing $p$. Using the lifting results of BL-G-G-T, we may lift this to a compatible family of self-dual representations of level one and weight zero which is potentially modular. Because these representations are potentially modular and are not CM, we  know that they are all irreducible by Blasius-Rogawski. We now specialize these representations to $p=5$, and because the Hodge-Tate weights are sufficiently small ($[0,1,2]$) and $\mathbf{Q}(\sqrt{-1})$ is also small, we can use results of Fontaine and Abrashkin to deduce that the corresponding $5$-adic representation is reducible, which is a contradiction. We thus deduce (using Khare-Wintenberger for the odd case) that there do not exist any irreducible finite flat group schemes $G$ of type $(p,p)$ over $\mathrm{Spec}(\mathbf{Z}[\sqrt{-1}])$ whose generic fibre admits descent data to $\mathbf{Q}$. This entire argument is really just a version of the Khare-Wintenberger proof of Serre’s conjecture  for $U(3)$. Unfortunately, one doesn’t quite have enough modularity lifting theorems at this point to deduce Serre’s conjecture completely for $U(3)$.

These arguments are quite general. For example, there should only exist finitely many even representations $\overline{\rho}: G_{\mathbf{Q}}: \rightarrow \mathrm{GL}_n(\overline{\mathbf{F}}_p)$ (whose image contains $\mathrm{SL}_n(\mathbf{F}_p)$) of fixed Serre weight and level  even when one ranges over all primes $p$, providing $n > 2$.

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