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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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