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On the semi-simplicity of the \(U_p\)-operator on modular forms. (English) Zbl 0902.11020

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 Section 5 gives some (unpublished) results, due to Abbes and Ullmo, concerning the discriminants of certain Hecke algebras.

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
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