Dual Eichler-Shimura maps for the modular curve
Juan Esteban Rodriguez Camargo (ENS de Lyon)
Abstract: Andreatta-Iovita-Stevens have constructed interpolations of the small slope part of the Eichler-Shimura decomposition for the modular curve. Roughly speaking, they defined in a geometric way a map from the overconvergent modular symbols of weight k, to the overconvergent modular forms of weight k+2. Then, using classicality theorems of Coleman and Ash-Stevens, they achieved a Hodge-Tate decomposition of the small slope part of overconvergent modular symbols. On the other hand, in a recent paper of Boxer-Pilloni, the authors proved that higher Coleman and Hida theories exist for the modular curve. The aim of this talk is to construct geometrically a map from the higher cohomology of overconvergent modular forms of weight -k to the modular symbols as above. We shall recover the Hodge-Tate decomposition of the small slope part of modular symbols, with the addition that all the maps involved are defined using the geometry of the modular curve. If time permits, we will discuss the compatibility of the previous work with duality.
number theory
Audience: researchers in the topic
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Organizers: | Caleb Springer*, Luis Garcia* |
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