The $${\mathcal {L}}$$ -invariant, the dual $${\mathcal {L}}$$ -invariant, and families
Tóm tắt
Given a rank two trianguline family of
$$(\varphi ,\Gamma )$$
-modules having a noncrystalline semistable member, we compute the Fontaine–Mazur
$${\mathcal {L}}$$
-invariant of that member in terms of the logarithmic derivative, with respect to the Sen weight, of the value at p of the trianguline parameter. This generalizes prior work, in the case of Galois representations, due to Greenberg–Stevens and Colmez.
Tài liệu tham khảo
Colmez, P.: Zéros supplémentaires de fonctions \(L\)-adiques de formes modularies. Algebra and number theory. Tandon, R. (ed.) Hindustan Book Agency, 193–210 (2005)
Colmez, P.: Invariants \({\cal L}\) et dérivées de valeurs propres de Frobenius. Astérisque 331, 13–28 (2010)
Greenberg, Ralph, Stevens, Glenn: \(p\)-Adic \(L\)-functions and \(p\)-adic periods of modular forms. Invent. Math. 111(2), 407–447 (1993)
Kedlaya, K.S., Pottharst, J., Xiao, L.: Cohomology of arithmetic families of \((\varphi ,\Gamma )\)-modules. J. Amer. Math. Soc. 27(4), 1043–1115 (2014)