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Another look at Gross-Stark units over the number field \(\mathbb Q\). (English) Zbl 1231.11135

Author’s summary: We provide another description of the Gross–Stark units over the rational field \({\mathbb Q}\) (studied in [B. Gross, J. Fac. Sci., Univ. Tokyo, Sect. I A 28, 979–994 (1981; Zbl 0507.12010)]), which is essentially a Gauss sum using a \(p\)-adic multiplicative integral of the \(p\)-adic Kubota–Leopold distribution and give a simplified proof of the Ferrero–Greenberg theorem see [B. Ferrero and R. Greenberg, Invent. Math. 50, 91–102 (1978; Zbl 0441.12003)] for \(p\)-adic Hurwitz zeta functions. This is a precise analog for \({\mathbb Q}\) of Darmon-Dasgupta’s work on elliptic units for real quadratic fields [H. Darmon and S. Dasgupta, Ann. Math. (2) 163, No. 1, 301–346 (2006; Zbl 1130.11030)].

MSC:

11S40 Zeta functions and \(L\)-functions
11G16 Elliptic and modular units
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References:

[1] DOI: 10.2307/3062142 · Zbl 1035.11027 · doi:10.2307/3062142
[2] DOI: 10.4007/annals.2006.163.301 · Zbl 1130.11030 · doi:10.4007/annals.2006.163.301
[3] DOI: 10.4153/CJM-2007-023-0 · Zbl 1118.11045 · doi:10.4153/CJM-2007-023-0
[4] DOI: 10.1215/00127094-2008-019 · Zbl 1235.11102 · doi:10.1215/00127094-2008-019
[5] DOI: 10.1007/BF01406470 · Zbl 0441.12003 · doi:10.1007/BF01406470
[6] Gross B., J. Fac. Sci. Univ. Tokyo 28 pp 979–
[7] DOI: 10.2307/1971226 · Zbl 0406.12010 · doi:10.2307/1971226
[8] DOI: 10.1007/BF01394274 · Zbl 0666.12009 · doi:10.1007/BF01394274
[9] DOI: 10.1017/CBO9780511526107 · doi:10.1017/CBO9780511526107
[10] DOI: 10.1007/978-1-4612-1934-7 · doi:10.1007/978-1-4612-1934-7
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