The local Serre weight conjecture at a ramified prime

Let $\mathfrak p\mid p$ be a place of $F$, with ramification index $e$, residue degree $s$, and inertia group $I_{\mathfrak p}$. Let $W_{\mathfrak p}^?(\rho)$ denote the predicted Serre weights at $\mathfrak p$. For $e\le p-1$, let $\Delta=[0,e-1]^I$, let $\mathcal R_{\mathfrak p}^{\delta}$ be the multi-valued operation on the set $Y_{\mathfrak p}$ of Serre weights at $\mathfrak p$, and let $JH(\overline{V_{\mathfrak p}(\rho)})$ be the set of Jordan–Hölder constituents of the reduction of the associated characteristic-zero representation. If $e\ge p$, write

$$F(a,b)=\bigotimes_{j\in\mathbb Z/s\mathbb Z}(\det^{w_j}\operatorname{Sym}^{k_j-2}k_{\mathfrak p}^2)\otimes_{k_{\mathfrak p},\tau_j}\overline{\mathbb F}_p,$$

where $b=\sum_{j=0}^{s-1}w_jp^{s-j}$ and $a-b=\sum_{j=0}^{s-1}(k_j-2)p^{s-j}$, with $0\le w_j\le p-1$ and $2\le k_j\le p+1$. Local Serre weight conjecture. If $\rho:\operatorname{Gal}(\overline F/F)\to\operatorname{GL}_2(\overline{\mathbb F}_p)$ is continuous, irreducible, totally odd, and tame at $\mathfrak p$, then

$$W_{\mathfrak p}^?(\rho)=\bigcup_{\delta\in\Delta}\mathcal R_{\mathfrak p}^{\delta}\bigl(JH(\overline{V_{\mathfrak p}(\rho)})\bigr)$$

if $e\le p-1$, while

$$W_{\mathfrak p}^?(\rho)=\left\{F(a,b):\det\rho\vert_{I_{\mathfrak p}}=\lambda_0^{a+b+\sum_{j=0}^{s-1}ep^j}\right\}$$

if $e\ge p$. The conjecture gives the predicted local weight set, with the large-ramification case asserting that every weight satisfying the determinant, or central-character, condition is modular. The paper proves some cases and motivates the maximal predicted set for $e\ge p$ by the corresponding behavior at $e=p-1$.