[SOLVED] Mathematics

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 / pmathcal{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
eq 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.