Lubin–Tate and multivariable (𝜑,𝒪_{𝒦}^{×})-modules in dimension 2

Source abstract

Let p p be a prime number, K K a finite unramified extension of Q p \mathbb {Q}_p and F \mathbb {F} a finite extension of F p \mathbb {F}_p . For ρ ¯ \overline {\rho } any reducible two-dimensional representation of G a l ( K ¯ / K ) Gal(\overline {K}/K) over F \mathbb {F} , we compute explicitly the associated étale ( φ , O K × ) (\varphi ,\mathcal {O}_K^{\times }) -module D A ⊗ ( ρ ¯ ) D_A^{\otimes }(\overline {\rho }) defined by Christophe Breuil, Florian Herzig, Yongquan Hu, Stefano Morra, and Benjamin Schraen [Math. Ann. 392 (2025), pp. 2709–2801]. Then we let π \pi be an admissible smooth representation of G L 2 ( K ) GL_2(K) over F \mathbb {F} occurring in some Hecke eigenspaces of the mod p p cohomology and ρ ¯ \overline {\rho } be its underlying two-dimensional representation of G a l ( K ¯ / K ) Gal(\overline {K}/K) over F \mathbb {F} . Assuming that ρ ¯ \overline {\rho } is maximally non-split, we prove under some genericity assumption that the associated étale ( φ , O K × ) (\varphi ,\mathcal {O}_K^{\times }) -module D A ( π ) D_A(\pi ) defined by Christophe Breuil, Florian Herzig, Yongquan Hu, Stefano Morra, and Benjamin Schraen [Mem. Amer. Math. Soc. 315 (2025), pp. v+163] is isomorphic to D A ⊗ ( ρ ¯ ) D_A^{\otimes }(\overline {\rho }) . This extends the results of Christophe Breuil, Florian Herzig, Yongquan Hu, Stefano Morra, and Benjamin Schraen [Math. Ann. 392 (2025), pp. 2709–2801], where ρ ¯ \overline {\rho } was assumed to be semisimple.