On the meromorphic continuation of degree two \(L\)-functions (Q2369743)

The author extends his earlier work [J. Inst. Math. Jussieu 1, No. 1, 125--143 (2002; Zbl 1047.11051)] from the ordinary to the crystalline, low weight case. This allows him to prove that the \(L\)-function of any regular (i.e. having distinct Hodge numbers), irreducible, rank two motive over \(\mathbb{Q}\) has meromorphic continuation to the whole complex plane and satisfies the expected functional equation (Theorem A). More precisely, he expresses any such \(L\)-function as a ratio of products of \(L\)-functions attached to certain Hilbert modular forms over different subfields of a certain totally real Galois extension of \(\mathbb{Q}\). The case of rigid Calabi-Yau \(3\)-fold is discussed in some detail in Section 6. He also proves a special case of the Fontaine-Mazur conjecture (Theorem B).