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Shintani cocycles and the order of vanishing of \(p\)-adic Hecke \(L\)-series at \(s=0\)

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Abstract

Let \(\chi \) be a Hecke character of finite order of a totally real number field \(F\). By using Hill’s Shintani cocycle we provide a cohomological construction of the \(p\)-adic \(L\)-series \(L_p(\chi , s)\) associated to \(\chi \). This is used to show that \(L_p(\chi , s)\) has a trivial zero at \(s=0\) of order at least equal to the number of places of \(F\) above \(p\) where the local component of \(\chi \) is trivial.

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Notes

  1. This normalization is not standard; usually this \(p\)-adic \(L\)-series is denote by \(L_p(\chi \omega ,s)\).

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  1. Fakultät für Mathematik, Universität Bielefeld, Bielefeld, Germany

    Michael Spiess

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  1. Michael Spiess
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Correspondence to Michael Spiess.

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Spiess, M. Shintani cocycles and the order of vanishing of \(p\)-adic Hecke \(L\)-series at \(s=0\) . Math. Ann. 359, 239–265 (2014). https://doi.org/10.1007/s00208-013-0983-5

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  • DOI: https://doi.org/10.1007/s00208-013-0983-5

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