On the properness of the eigencurve associated to unitary Shimura curves

• Main Author: Chu, Simon
• Other Authors: Diamond, Fred Irvin ; L. Kassaei, Payman
• Published: King's College London (University of London) 2016
• Subjects: 516.3
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.700791

We study overconvergent modular forms on certain unitary Shimura curves, dened for integral weights by Kassaei and general weights by Brasca. There are Hecke operators acting on these spaces of overconvergent modular forms, and there is a distinguished UP-operator which is a compact operator on these spaces. We construct a deformationtheoretic eigencurve in this setting, which comes with a projection to the weight space. Then we prove that its nilreduction is isomorphic to the Hecke eigencurve. In particular, for each weight in the weight space, the bre above it is idented with systems of Hecke eigenvalues arising from overconvergent eigenforms of that weight, whose UPeigenvalu e is not 0. Lastly, we prove that this eigencurve is proper (that is, the map to the weight space satises the valuative criterion of properness). This is done using padic Hodge theory, via interpreting the UPeigenvalues on the automorphic side as the eigenvalues of Frobenius on the padic Hodge theory side, for families of Galois representations attached to nite slope, overconvergent eigenforms.