The epsilon constant conjecture for higher dimensional unramified twists of \(\mathbb{Z}_p^r(1)\). (English) Zbl 1511.11095
Let \(p\) be a prime integer and \(N/K\) a finite Galois extension of \(p\)-adic number fields with group \(G:=\mathrm{Gal}(N/K)\). Denote the absolute Galois group of \(K\) by \(G_{K}\). Let \(\rho^{nr}:G_{K}\to Gl_{r}(Z_{p})\) be an \(r\)-dimensional unramified representation of the absolute Galois group \(G_{K}\) which is the restriction of an unramified representation and \(\rho_{Q_{p}}^{nr}: G_{Q_{p}}\to Gl_{r}(Z_{p})\). The authors of the paper under review generalize previous work of D. Izychev and O. Venjakob [J. Théor. Nombres Bordx. 28, No. 2, 485–521 (2016; Zbl 1411.11105)] in the tame case and of the authors in the weakly and wildly ramified case to prove the conjecture for all tame extensions \(N/K\) and a certain family of weakly and wildly ramified extensions \(N/K\). The paper is a lengthy well written and is an excellent source for interested researchers in the field.
Reviewer: Manouchehr Misaghian (Prairie View)
MSC:
| 11S23 | Integral representations |
| 11S25 | Galois cohomology |
| 11S40 | Zeta functions and \(L\)-functions |
Keywords:
epsilon constant conjecture; equivariant Tamagawa number conjecture; weakly ramified extensionsCitations:
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