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\(\mathcal L\) invariant and \(p\)-adic special series. (Invariant \(\mathcal L\) et série spéciale \(p\)-adique.) (French) Zbl 1166.11331

Summary: To any integer \(k\geq 2\) and any \(\mathcal L\in\overline{\mathbb Q_p}\), we associate conjecturally a \(p\)-adic Banach space \(B(k,L)\) endowed with a continuous linear action of \(\text{GL}_2(\mathbb Q_p)\). We prove that \(B(k,L)\) does exist either if \(k=2\) or if \(k>2\) and \(L\) is “coming from” a weight \(k\) eigenform on \(\Gamma_0(pN)\) with \((p,N)=1\) and \(N=N^-N^+\) where \((N^-,N^+)=1\) and \(N^-\) is the product of an odd number of prime numbers. The Banach space \(B(k,L)\) should “correspond” (up to torsion by crystalline characters) to 2-dimensional non-crystalline semi-stable representations of \(\text{Gal} (\overline{\mathbb Q_p}/\mathbb Q_p)\) over \(\overline{\mathbb Q_p}\).

MSC:

11F85 \(p\)-adic theory, local fields
11F80 Galois representations
22E50 Representations of Lie and linear algebraic groups over local fields
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