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Shintani cocycles and the order of vanishing of $$p$$-adic Hecke $$L$$-series at $$s=0$$. (English) Zbl 1307.11125
Let $$F\neq Q$$ be a totally real number field, and let $$\chi$$ be a totally odd Hecke character of $$F$$, having finite order. For a rational prime $$p$$ let $$r$$ be the number of prime ideals in $$F$$ lying over $$p$$, at which the local component of $$\chi$$ equals $$1$$, and let $$L_p(\chi,s)$$ be the $$p$$-adic $$L$$-function attached to $$\chi$$. The main result of the paper is a new proof of the inequality $\text{ord}_{s=0}L_p(\chi,s)\geq r,$ which is a consequence of Iwasawa’s Main Conjecture, established by A. Wiles [Ann. Math. (2) 131, No. 3, 493–540 (1990; Zbl 0719.11071)]. Another proof has been given by P. Charollois and S. Dasgupta [Camb. J. Math. 2, No. 1, 49–90 (2014; Zbl 1353.11074)]. The author’s proof is based on his previous paper [Invent. Math. 196, No. 1, 69–138 (2014; Zbl 1392.11027)] in which he considered the same question for $$p$$-adic $$L$$-functions of elliptic curves over totally real fields.

##### MSC:
 11S40 Zeta functions and $$L$$-functions 11R23 Iwasawa theory 11R80 Totally real fields
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##### References:
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