Let p be an odd prime, K a finite extension of \mathbb{Q}_p with ring of integers \mathcal{O}_K and residue field k = \mathcal{O}_K / p\mathcal{O}_K (of characteristic p ). Let X be a proper smooth algebraic variety over \mathcal{O}_K , whose special fiber X_k = X \times_{\mathcal{O}_K} k is geometrically connected, and whose generic fiber X_K = X \times_{\mathcal{O}_K} K is an abelian variety with complex multiplication (CM).
Let \ell \neq p be another prime, and let \rho: \text{Gal}(\overline{K}/K) \to \text{GL}_{2g}(\mathbb{Q}_\ell) be the Galois representation induced by the \ell -adic Tate module of X_K , where g = \dim X_K . Assume \rho is semisimple and its image is contained in a split torus (i.e., a "potentially abelian" representation).
1. Prove that there exists a crystalline representation \rho_{\text{cr}} with Hodge-Tate weights \{0, 1, \dots, 2g-1\} associated to \rho , and that its Fontaine-Laffaille module satisfies specific filtration conditions.
2. If X_k is supersingular, show that the eigenvalues of the Hecke algebra action on H^1_{\text{et}}(X_{\overline{K}}, \mathbb{Q}_\ell) are in bijection with embeddings of elements of some ring of integers of the CM field of X into \mathbb{Q}_\ell .
3. Using the above results, explain how this Galois representation satisfies the local case of the Langlands program's conjecture relating Galois representations of abelian varieties to automorphic forms over non-archimedean local fields.
This problem lies at the intersection of arithmetic algebraic geometry, p-adic Hodge theory, and the Langlands program—core areas of modern research in number theory. It requires mastery of:
- Fontaine's rings ( B_{\text{cr}}, B_{\text{st}} ) and the classification of crystalline representations;
- The arithmetic of CM abelian varieties and the Galois action on their Tate modules;
- Deep connections between Galois representations and automorphic forms, particularly in local settings.