Connection between the $L$-function of a modular form and the $L$-function of its associated $\ell$-adic representation as defined by Bloch-Kato

Question

Let $f\in\mathcal{S}_k(\Gamma_1(N),\epsilon)$ is a normalized eigenform for all the Hecke operators, with character $\epsilon$, $k\geq 2$, and assume the $q$-expansion of $f$ has rational coefficients. The $L$-function $L(f,s)$ admits an Euler product of the form

$L(f,s)=\prod_{p\text{ prime}}(1-a_pp^{-s}+\epsilon(p)p^{k-1-2s})^{-1}$.

On the other hand, for a fixed prime $\ell$, we have the associated $\ell$-adic Galois representation $\rho_f:G_\mathbf{Q}\rightarrow\mathrm{GL}_2(\mathbf{Q}_\ell)$ unramified outside $\ell N$ with the characteristic polynomial of $\rho_f$ of an arithmetic Frobenius at $p\nmid\ell N$ given by $X^2-a_pX+\epsilon(p)p^{k-1}$. Bloch and Kato have defined local $L$-factors for $\rho_f$ at each prime $p$. If $p\neq\ell$, they define the local $L$-factor to be the reciprocal of the polynomial $\det(I-\rho_f(\mathrm{Frob}_p^{-1})X\vert \rho_f^{I_p})$ evaluated at $p^{-s}$, where $I_p$ is an inertia group at $p$. For $p=\ell$, the definition is similar but defined in terms of the Frobenius action of $D_{cris,\ell}(\rho_f)$.

I'm having trouble matching up the local $L$-factors for the $L$-function of $\rho_f$ with those of $f$, and I now wonder whether they coincide at all. If, for example, $p\nmid\ell N$, then, as matrices with coefficients in $\mathbf{Q}_p[X]$, we have

$I-\rho_f(\mathrm{Frob}_p^{-1})X=(-\rho_f(\mathrm{Frob}_p^{-1}))(IX-\rho_f(\mathrm{Frob}_p))$,

so, knowing what I do about the characteristic polynomial of $\rho_f(\mathrm{Frob}_p)$, I get that the Bloch-Kato local $L$-factor is $((\epsilon(p)^{-1}p^{1-k})(p^{-2s}-a_pp^{-s}+\epsilon(p)p^{k-1}))^{-1}$, which is $(1-a_pp^{1-k-s}+\epsilon(p)^{-1}p^{1-k-2s})^{-1}$. This (apparently) differs from the local $L$-factor at $p$ for $L(f,s)$.

My questions are:

(1) Have I gotten something wrong above?

(2) Should I even expect the $L$-function of the modular form to coincide with the $L$-function of the associated $\ell$-adic representation (maybe I need to dualize the representation or somehow normalize differently)?

Answer 1

Your definition of $\rho_{f,\ell}$ is out by a dualization, I think.

The normalizations in Bloch--Kato are such that you get the usual $L$-function of $f$ out if you put the Deligne representation $\rho_{f, \ell}$ in; but Deligne's normalization for $\rho_{f, \ell}$ is such that the characteristic polynomial of geometric Frobenius at $p \nmid \ell N$ is the Hecke polynomial (in particular, the determinant of $\rho_{f, \ell}$ is a finite-order character times the $(1-k)$-th power of the cyclotomic character, not the $(k-1)$-th). For example, for a weight 2 form attached to an elliptic curve, Deligne's representation is $H^1_{et}$ of the elliptic curve, which is the dual of the Tate module of the curve. (This is sometimes called the "cohomological" normalization, as opposed to the "homological" normalization which is what you're using.)