The equivariant Tamagawa number conjecture and the extended abelian Stark conjecture. (English) Zbl 1396.11125

Vallières, Daniel

J. Reine Angew. Math. 734, 1-58 (2018).

Author’s abstract: “The goal of this paper is to show that the equivariant Tamagawa number conjecture implies the extended abelian Stark conjecture contained in [S. Erickson, Rocky Mt. J. Math. 39, No. 3, 765–787 (2009; Zbl 1183.11071)] and [C. J. Emmons and C. D. Popescu, J. Number Theory 129, No. 6, 1350–1365 (2009; Zbl 1166.11043)]. In particular, this gives the first proof of the extended abelian Stark conjecture for the base field \(\mathbb{Q}\), since the equivariant Tamagawa number conjecture away from 2 was proved in this context by D. Burns and C. Greither [Invent. Math. 153, No. 2, 303–359 (2003; Zbl 1142.11076)] and M. Flach completed their results at 2 in [Contemp. Math. 358, 79–125 (2004; Zbl 1070.11025)] and [J. Reine Angew. Math. 661, 1–36 (2011; Zbl 1242.11083)].”
From the introduction: “The paper is subdivided as follows: In Section 2, we review Tate sequences, and in Secton 3 we state the equivariant Tamagawa number conjecture as formulated by D. Burns and M. Flach [Doc. Math. 6, 501–570 (2001; Zbl 1052.11077)]. Particularly important forms is the reformulation of the equivariant Tamagawa number conjecture contained in Section 3.4 which is due to Burns. In Section 4, we recall the statement of the Emmons-Popescu conjecture and we state a stronger conjecture (Conjecture 4.16) which will be the main object at study of this paper. We also study the connection between this stronger conjecture and a question that was studied in [J. Tate, Progress in Mathematics, Vol. 47. Boston-Basel-Stuttgart: Birkhäuser. (1984; Zbl 0545.12009)] and [the author, J. Number Theory 132, No. 11, 2535–2567 (2012; Zbl 1368.11117)]. In Section 5, we present a useful reduction. We point out here that even if one is only interested in the rank one situation, this reduction forces the consideration of the higher-order of vanishing case as well. We go on in Section 6 with showing that the equivariant Tamagawa number conjecture implies Conjecture 4.16 following the strategy used by D. Burns [Invent. Math. 169, No. 3, 451–499 (2007; Zbl 1133.11063)] in the case of the Rubin-Stark conjecture. This is our main theorem which we state in Section 6.3. We end this paper with Section 7, where we state some explicit consequences of our work when the base field is \(\mathbb{Q}\).”

Reviewer: Andrzej Dąbrowski (Szczecin)

MSC:

11R23	Iwasawa theory
11G40	\(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture

Keywords:

equivariant Tamagawa number conjecture; abelian Stark conjecture; Tate sequences; Rubin-Stark conjecture

Citations:

Zbl 1183.11071; Zbl 1166.11043; Zbl 1142.11076; Zbl 1070.11025; Zbl 1242.11083; Zbl 1052.11077; Zbl 0545.12009; Zbl 1368.11117; Zbl 1133.11063

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