The zig-zag conjecture for reductions of crystalline representations

Let $p$ be an odd prime, let $V_{k,a_p}$ be a two-dimensional crystalline representation of $\operatorname{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$, and let $v_p(a_p)$ be a half-integral slope satisfying

$$\frac{1}{2}\leq v_p(a_p)\leq\frac{p-1}{2}.$$

Let $k_0=2v_p(a_p)+2$ modulo $p-1$ be the corresponding exceptional-weight congruence class, set $r=k-2$ and $r_0=k_0-2$, and define

$$\tau=v_p(c),\qquad t=v_p(k-k_0),$$

where

$$c=\frac{a_p^2-\binom{r-v_-}{v_+}\binom{r-v_+}{v_-}p^{r_0}}{pa_p},$$

with $v_-$ and $v_+$ the largest and smallest integers such that $v_p(a_p)\in(v_-,v_+)$. Zig-zag conjecture. For all sufficiently large weights $k\geq k_0$ with $t$ sufficiently large, the semisimplification $\bar V_{k,a_p}$ of the reduction of $V_{k,a_p}$ is

$$\bar V_{k,a_p}\sim\begin{cases}\operatorname{ind}(\omega_2^{r_0+1+i(p-1)})&\text{if }\tau\in(t+i-1,t+i),\\[2pt]\mu_{\lambda_i}\cdot\omega^{r_0-i}\oplus\mu_{\lambda_i^{-1}}\cdot\omega^{1+i}&\text{if }\tau=t+i, \end{cases}$$

for $0\leq i\leq(r_0-1)/2$ when $r_0$ is odd and $0\leq i\leq r_0/2$ when $r_0$ is even, with constants $\lambda_i\in\overline{\mathbb{F}}_p^*$. Thus, as $\tau$ varies through the $t$-line, the reductions alternate between irreducible and reducible representations. This conjecture describes the exceptional cases not covered by the Breuil–Buzzard–Emerton conjecture; the paper proves it in families and establishes several cases, while the full pattern and the precise constants remain the subject of the stated conjectural formulation.