Dear MO,
This is a follow up to a previous question here in MO, but I will make this question self-contained for convenience. Those already familiar with the following paper [G] by Gross can safely skip down to the questions below.
In [G],
- Benedict H. Gross, "A tameness criterion for Galois representations associated to modular forms (mod $p$)", Duke Math. Journal, Vol. 61, No. 2, 1990,
Gross shows criteria (A) and (B) below. Let $p$ be a prime, and let $f=\sum a_nq^n$ be a normalized cuspidal eigenform of weight $k$ and character $\varepsilon$ for $\Gamma_1(N)$, with coefficients in a finite field $\mathbb{F}$ of characteristic $p$. Let $$\rho_f:\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to \operatorname{GL}_2(\mathbb{F})$$ be a continuous semi-simple Galois representation such that $\rho_f$ is unramified for all primes $\ell$ such that $\gcd(\ell,Np)=1$, and the matrix $\rho_f(Frob_\ell)$ has characteristic polynomial $x^2-a_\ell x+\varepsilon(\ell)\ell^{k-1}$.
Assume $2\leq k\leq p$ and $a_p\neq 0$ (if $k=p$, assume $a_p^2\neq \varepsilon(p)$. The criterion says as follows:
- (A) The representation $\rho_f$ is completely reducible when restricted to a decomposition group at $p$ if and only if there is a companion form for $f$, i.e., there is $g=\sum b_nq^n$ of weight $k'=p+1-k$ and character $\varepsilon$ for $\Gamma_1(N)$ over $\mathbb{F}$, such that $n^kb_n=na_n$, for all $n\geq 1$.
The statements with all the details are in [G], Proposition 13.8 and Theorem 13.10.
Now jump to p. 514 of [G]. Suppose that $f$ is the reduction mod $p$ of a newform $F$ of weight $2$, where $F$ has trivial character on $\Gamma_1(N)$, and has coefficients in $\mathbb{Z}$. Then, $F$ corresponds to an elliptic curve $E/\mathbb{Q}$ of conductor $N$. Let $$\rho_{E,p} : \operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \operatorname{GL}_2(\mathbb{Z}_p)$$ be the representation associated to the action of Galois on the Tate module $T_p(E)$. Let us denote $\rho_{E,p} \bmod p$ by ${\overline{\rho}}$. Then Gross deduces the following criterion, when $E$ has good ordinary reduction at $p$, with $j(E)\neq 0, 1728$, and $\overline{\rho}$ is irreducible:
- (B) The restriction of $\rho_{E,p} \bmod p^n$ to a decomposition group at $p$ is diagonalizable if and only if $j(E)\equiv j_0 \bmod 2p^{n+1}$, where $j_0$ is the $j$-invariant of the "canonical lifting" to $\mathbb{Q}_p$ of $E \bmod p$.
Now my questions:
Question 1: I take it that Gross assumes that $\overline{\rho}$ be irreducible in (B) so it coincides with its semi-simplification, and therefore $\overline{\rho}$ is equivalent to $\rho_f$ and (A) applies. But, does criterion (B) still hold if $\overline{\rho}$ is reducible? I.e., what if $E$ has a $\mathbb{Q}$-rational $p$-isogeny? For an example, see my previous question here in MO, where I specify my interest in the curve $E$ with Cremona label 1225h1, which has a rational $37$-isogeny.
Question 2: If (B) does not hold when $\overline{\rho}$ is reducible (or if it is not known whether it holds), do any of the two directions of the if-and-only-if hold when $\overline{\rho}$ is reducible?
Question 3: If $E/\mathbb{Q}$ has potential good ordinary reduction, does (B) still hold?
Thank you!
