×

\(\mathfrak p\)-adic modular forms over Shimura curves over totally real fields. (English) Zbl 1052.11037

The theory of \(p\)-adic modular forms was initiated by J.-P. Serre [Lect. Notes Math. 350, 191–268 (1973; Zbl 0277.12014)]. He defined \(p\)-adic modular forms as \(p\)-adic limits of classical modular forms of varying weights. N. M. Katz [Lect. Notes Math. 350, 69–190 (1973; Zbl 0271.10033)] later gave a geometric interpretation: he defined \(p\)-adic modular forms (of integral weight) as certain functions on the moduli space of test objects consisting of an ordinary elliptic curve with a level structure, and gave a modular interpretation of the action of Hecke operators (including the \(\mathcal U\) operator). Let \(F\) be a totally real field. Let \({\mathfrak p}\) be a prime ideal of the ring of integers \({\mathcal O}_F\), which lies over a fixed rational prime \(p\). Let \({\mathcal O}_{\mathfrak p}\) denote the completion of \({\mathcal O}_F\) at \({\mathfrak p}\), and let \(R_0\supset{\mathcal O}_{\mathfrak p}\) be a complete discrete valuation ring, with valuation \(v\), such that \(v(\pi)=1\) (where \(\pi\) is a uniformizer of \({\mathcal O}_{\mathfrak p}\). The author defines, following Katz, the spaces of \({\mathfrak p}\)-adic modular forms of growth condition \(r\in R_0\) over certain unitary Shimura curves (§8). He also describes, inspired by R. Coleman’s method [Invent. Math. 124, No. 1-3, 215–241 (1996; Zbl 0851.11030)], \({\mathfrak p}\)-adic modular forms for more general ground rings using rigid analytic varieties and formal schemes. He defines the \(\mathcal U\) operator in a such general context (essentially) as a trace of Frob (the Frobenius morphism of \({\mathfrak p}\)-adic modular forms), proves, that the space of overconvergent modular forms (defined in §1, p. 361) is invariant under \(\mathcal U\), and studies the overconvergent eigenforms of \(\mathcal U\). He also shows that \(\mathcal U\) is a completely continuous operator when \(r\) is not a unit in \(R_0\).

MSC:

11F85 \(p\)-adic theory, local fields
11F55 Other groups and their modular and automorphic forms (several variables)
11F33 Congruences for modular and \(p\)-adic modular forms
11G35 Varieties over global fields
PDFBibTeX XMLCite
Full Text: DOI