Algebra and Number Theory Seminar: Jacob Swenberg, UCLA

A Λ-adic Eichler-Shimura isomorphism for Shimura curves

Abstract: In a series of papers, Ohta proved that the projective limit of ordinary étale cohomology groups of the modular curves 𝑋1⁡(𝑁⁢𝑝𝑟) has a natural quotient isomorphic to Hida's space of Λ-adic cusp forms. We generalize these results to Shimura curves over an arbitrary totally real number field 𝐹. Let 𝐵 be a quaternion algebra over 𝐹 split at exactly one real place, 𝔑 ⊆O𝐹 an ideal and 𝔭 a prime of 𝐹 of odd residue characteristic 𝑝 such that 𝐵 is split at all primes dividing 𝔑⁢𝔭. Let Λ =lim⟵𝑟⁡ℤ𝑝⁢[𝐶⁢𝑙+𝐹⁡(𝔑⁢𝔭𝑟)] and 𝑋𝑟 =𝑋𝐵1⁡(𝔑⁢𝔭𝑟). We prove that the 𝑇⁢(𝔭)∗-ordinary part of lim⟵𝑟⁡𝐻1ét⁡(𝑋𝑟,――𝐹,ℤ𝑝) has a Λ⁡[𝐺𝐹𝔭]-module quotient which is isomorphic to a projective limit of integral differentials. We also prove a Λ-adic duality theorem and discuss relevant control theorems.