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On the quaternionic $$p$$-adic $$L$$-functions associated to Hilbert modular eigenforms. (English) Zbl 1253.11088
Int. J. Number Theory 8, No. 4, 1005-1039 (2012); erratum ibid. 12, No. 1, 305-311 (2016).
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 M. Bertolini and 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 H. Darmon and A. Iovita [J. Inst. Math. Jussieu 7, No. 2, 291–325 (2008; Zbl 1146.11057)] and 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 X. Yuan, S. Zhang and W. Zhang [Heights of CM points. I: Gross-Zagier formula, preprint, to appear in Annals of Mathematical Studies, Princeton University Press].
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 V. Vatsal [Duke Math. J. 116, No. 2, 219–261 (2003; Zbl 1065.11048)].
The last section contains a conjectural non-vanishing criterion of 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).

##### MSC:
 11M38 Zeta and $$L$$-functions in characteristic $$p$$ 11F41 Automorphic forms on $$\mbox{GL}(2)$$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11R23 Iwasawa theory
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