Smoothness and classicality on eigenvarieties. (English) Zbl 1410.11042
“Let \(p\) be a prime number. In this paper we are concerned with classicality of \(p\)-adic automorphic forms on some unitary groups, i.e. we are looking for criteria that decide whether a given \(p\)-adic automorphic form is classical or not. More precisely, we work with \(p\)-adic forms of finite slope, that is, in the context of eigenvarieties.”
“Let \(F^+\) be a totally real number field and \(F\) be an imaginary quadratic extension of \(F^+\). We fix a unitary group \(G\) in \(n\) variables over \(F^+\) which splits over \(F\) and over all \(p\)-adic places of \(F^+\), and which is compact at all infinite places of \(F^+\).”
One may view a \(p\)-adic overconvergent eigenform as a point \(x\) of the associated Hecke eigenvariety, and one can associate to each such \(x\) a continuous semi-simple representation \(\varphi_X: \text{Gal}(\overline F/F)\to \text{GL}_n(\overline{\mathbb{Q}}_p)\) which is unramified outside a finite set of places of \(F\) and trianguline at all places of \(F\) dividing \(p\).
One expects that any overconvergence form \(x\) of classical weight such that \(\rho_x\) is de Rham at places of \(F\) dividing \(p\) is a classical automorphic form (Conjecture 3.5).
“Such a classicality theorem is due to Kisin in the context of Carleman-Mazur’s eigencurve, i.e. in the slightly different setting \(\text{GL}_1/\mathbb{Q}\).”
“In present paper, we prove new cases of this classicality conjecture (in the above unitary setting). In particular we are able to deal with cases where the overconvergent form \(x\) is critical.”
“Let \(F^+\) be a totally real number field and \(F\) be an imaginary quadratic extension of \(F^+\). We fix a unitary group \(G\) in \(n\) variables over \(F^+\) which splits over \(F\) and over all \(p\)-adic places of \(F^+\), and which is compact at all infinite places of \(F^+\).”
One may view a \(p\)-adic overconvergent eigenform as a point \(x\) of the associated Hecke eigenvariety, and one can associate to each such \(x\) a continuous semi-simple representation \(\varphi_X: \text{Gal}(\overline F/F)\to \text{GL}_n(\overline{\mathbb{Q}}_p)\) which is unramified outside a finite set of places of \(F\) and trianguline at all places of \(F\) dividing \(p\).
One expects that any overconvergence form \(x\) of classical weight such that \(\rho_x\) is de Rham at places of \(F\) dividing \(p\) is a classical automorphic form (Conjecture 3.5).
“Such a classicality theorem is due to Kisin in the context of Carleman-Mazur’s eigencurve, i.e. in the slightly different setting \(\text{GL}_1/\mathbb{Q}\).”
“In present paper, we prove new cases of this classicality conjecture (in the above unitary setting). In particular we are able to deal with cases where the overconvergent form \(x\) is critical.”
Reviewer: Andrzej Dąbrowski (Szczecin)
MSC:
| 11F80 | Galois representations |
| 11F85 | \(p\)-adic theory, local fields |
| 11S31 | Class field theory; \(p\)-adic formal groups |
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