Overconvergent Modular Forms and Perfectoid Shimura Curves

  • Przemysław Chojecki

    Instytut Matematyczny PAN, ul. Sniadeckich 8, 00-656 Warszawa, Poland
  • D. Hansen

    Department of Mathematics, Columbia University, 2990 Broadway New York NY 10027
  • C. Johansson

    DPMMS, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, United Kingdom
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We give a new construction of overconvergent modular forms of arbitrary weights, defining them in terms of functions on certain affinoid subsets of Scholze's infinite-level modular curve. These affinoid subsets, and a certain canonical coordinate on them, play a role in our construction which is strongly analogous with the role of the upper half-plane and its coordinate 'zz' in the classical analytic theory of modular forms. As one application of these ideas, we define and study an overconvergent Eichler-Shimura map in the context of compact Shimura curves over Q\Bbb{Q}, proving stronger analogues of results of Andreatta-Iovita-Stevens.

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Przemysław Chojecki, D. Hansen, C. Johansson, Overconvergent Modular Forms and Perfectoid Shimura Curves. Doc. Math. 22 (2017), pp. 191–262

DOI 10.4171/DM/564