On the quaternionic \(p\)-adic \(L\)-functions associated to Hilbert modular eigenforms
The author constructs \(p\)-adic \(L\)-functions associated to cuspidal Hilbert modular eigenforms of parallel weight two in certain dihedral or anticyclotomic extensions. The construction generalizes those of \textit{M. Bertolini} and \textit{H. Darmon} [Invent. Math. 126, No. 3, 413--456 (1996; Zbl 0882.11034); Ann. Math. (2) 162, No. 1, 1--64 (2005; Zbl 1093.11037)] in the ordinary case, as well as constructions of \textit{H. Darmon} and \textit{A. Iovita} [J. Inst. Math. Jussieu 7, No. 2, 291--325 (2008; Zbl 1146.11057)] and \textit{R. Pollack} [Duke Math. J. 118, No. 3, 523--558 (2003; Zbl 1074.11061)] in the supersingular case. The proof uses the refinement of Waldspurger's theorem, given by \textit{X. Yuan}, \textit{S. Zhang} and \textit{W. Zhang} [Heights of CM points. I: Gross-Zagier formula, preprint, to appear in Annals of Mathematical Studies, Princeton University Press].NEWLINENEWLINE The author also gives an expression for the Iwasawa \(p\)-invariant associated to the constructed \(p\)-adic \(L\)-functions (Theorem 4.14) following the method of \textit{V. Vatsal} [Duke Math. J. 116, No. 2, 219--261 (2003; Zbl 1065.11048)].NEWLINENEWLINE The last section contains a conjectural non-vanishing criterion of \textit{B. Howard} type for these \(p\)-adic \(L\)-functions (compare [J. Reine Angew. Math. 597, 1--25 (2006; Zbl 1127.11072); Theorem 3.2.3(c)]. This criterion, if satisfied, can be used to reduce the associated Iwasawa main conjecture to a certain non-triviality criterion for families of \(p\)-adic \(L\)-functions (Lemma 5.3).
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