Coleman, Robert F.; Edixhoven, Bas On the semi-simplicity of the \(U_p\)-operator on modular forms. (English) Zbl 0902.11020 Math. Ann. 310, No. 1, 119-127 (1998). Let \(S(N,k)\) denote the \(\mathbb{C}\)-vector space of cusp forms of level \(N\) and weight \(k\geq 2\). For a prime number \(p\mid N\), let \(U_p\) denote the Hecke operator acting on \(S(N,k)\). The authors prove that if \(k=2\) and \(p^3\) does not divide \(N\), then the operator \(U_p\) is semi-simple (Theorems 2.1 and 4.2). For weight \(k\geq 3\), they prove the same result under the assumption that certain crystalline Frobenius elements are semisimple (Theorem 4.2). Let us mention that the case \(k=1\) is completely different (see the introduction). Section 5 gives some (unpublished) results, due to Abbes and Ullmo, concerning the discriminants of certain Hecke algebras. Reviewer: A.Dabrowski (Szczecin) Cited in 3 ReviewsCited in 30 Documents MSC: 11F33 Congruences for modular and \(p\)-adic modular forms 14G20 Local ground fields in algebraic geometry 14G40 Arithmetic varieties and schemes; Arakelov theory; heights Keywords:semi-simplicity of the \(U_p\)-operator on modular forms; cusp forms; Hecke operator; crystalline Frobenius elements PDFBibTeX XMLCite \textit{R. F. Coleman} and \textit{B. Edixhoven}, Math. Ann. 310, No. 1, 119--127 (1998; Zbl 0902.11020) Full Text: DOI arXiv