Andreatta, Fabrizio; Iovita, Adrian [Urban, Eric] Triple product \(p\)-adic \(L\)-functions associated to finite slope \(p\)-adic families of modular forms. (English) Zbl 1494.11046 Duke Math. J. 170, No. 9, 1989-2083 (2021). It is known for a while how to attach a \(p\)-adic \(L\)-function to a triple of Hida families (or ordinary \(p\)-adic families). However the definition of such \(p\)-adic \(L\)-function does not make sense in the case of finite slope \(p\)-adic families. This paper introduces a new method to handle the case of finite slope families.One views the finite slope families of modular forms as overconvergent sections of the modular sheaves. The construction is a geometric one. Indeed this is an application of a very general geometric construction that the authors developed, called the vector bundles with marked sections.In an earlier paper [Contrib. Math. Comput. Sci. 7, 401–441 (2014; Zbl 1328.11052)], E. Urban had claimed a construction for the finite slope families. However this is a gap in that paper. In an appendix of this paper, Urban fixed the gap using results of this paper. Reviewer: Zhengyu Mao (Newark) Cited in 14 Documents MSC: 11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations 11F85 \(p\)-adic theory, local fields Keywords:de Rham cohomology; overconvergent modular forms; overconvergent modular sheaves; \(p\)-adic \(L\)-functions; vector bundles with marked sections Citations:Zbl 1328.11052 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] F. Andreatta and A. Iovita, Overconvergent de Rham Eichler-Shimura morphisms, preprint, arXiv:2006.02185v1 [math.NT]. [2] F. Andreatta and A. Iovita, Triple product p-adic L-functions asociated to finite slope p-adic families of modular forms, preprint, arXiv:1708.02785v2 [math.NT]. [3] F. Andreatta, A. Iovita, and V. Pilloni, p-adic families of Siegel modular cuspforms, Ann. of Math. 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