Conjectures of Brumer, Gross and Stark. (English) Zbl 1469.11427
Baake, Michael (ed.) et al., Spectral structures and topological methods in mathematics. Zürich: European Mathematical Society (EMS). EMS Ser. Congr. Rep., 365-388 (2019).
Let \(L/K\) be a finite Galois extension with Galois group \(G\). The values at \(s=0\) of the Artin \(L\)-series for the irreducible characters of \(G\) can be used to construct a Stickelberger element \(\theta_{L/K}\) in the center of \(\mathbb C[G]\). When \(G\) is abelian, a conjecture of Brumer predicts that \(\theta_{L/K}\) can be used to construct annihilators of the ideal class group of \(L\), and the refinement by Stark predicts the generators of the resulting principal ideals.
The present survey discusses extensions of these ideas to non-abelian \(G\) due to the author [Ann. Inst. Fourier 61, No. 6, 2577–2608 (2011; Zbl 1246.11176)] and D. Burns [Invent. Math. 186, No. 2, 291–371 (2011; Zbl 1239.11128)]. These are then related to the equivariant Tamagawa number conjecture, the equivariant main conjecture of Iwasawa theory, and the conjecture of B. Gross on the order of vanishing of \(p\)-adic \(L\)-functions at \(s=0\).
For the entire collection see [Zbl 1416.60012].
The present survey discusses extensions of these ideas to non-abelian \(G\) due to the author [Ann. Inst. Fourier 61, No. 6, 2577–2608 (2011; Zbl 1246.11176)] and D. Burns [Invent. Math. 186, No. 2, 291–371 (2011; Zbl 1239.11128)]. These are then related to the equivariant Tamagawa number conjecture, the equivariant main conjecture of Iwasawa theory, and the conjecture of B. Gross on the order of vanishing of \(p\)-adic \(L\)-functions at \(s=0\).
For the entire collection see [Zbl 1416.60012].
Reviewer: Lawrence C. Washington (College Park)
MSC:
| 11R23 | Iwasawa theory |
References:
| [1] | D. Barsky, Fonctions zetap-adiques d’une classe de rayon des corps de nombres totalement réels. InGroupe d’Etude d’Analyse Ultramétrique (5e année: 1977/78)(Y. Amice, D. Barsky, and P. Robba, eds.), Secrétariat Math., Paris, 1978, Exp. No. 16, pp. 1-23. · Zbl 0406.12008 |
| [2] | D. Burns, Equivariant Tamagawa numbers and Galois module theory. I,Compos. Math.129 (2001), 203-237. · Zbl 1014.11070 |
| [3] | D. Burns, On derivatives of ArtinL-series,Invent. Math.186 (2011), 291-371. · Zbl 1239.11128 |
| [4] | D. Burns, On derivatives ofp-adicL-series atsD0,J. Reine Angew. Math. (Crelle)(to appear). |
| [5] | D. Burns and M. Breuning, Leading terms of ArtinL-functions atsD0andsD1,Compos. Math.143 (2007), 1427-1464. · Zbl 1135.11060 |
| [6] | D. Burns and M. Flach, Tamagawa numbers for motives with (non-commutative) coefficients, Doc. Math.6 (2001), 501-570. · Zbl 1052.11077 |
| [7] | D. Burns and M. Flach, Tamagawa numbers for motives with (non-commutative) coefficients. II,Amer. J. Math.125 (2003), 475-512. · Zbl 1147.11317 |
| [8] | P. Cassou-Nogues, Valeurs aux entiers négatifs des fonctions zêta et fonctions zêtap-adiques, Invent. Math.51 (1979), 29-59. · Zbl 0408.12015 |
| [9] | T. Chinburg, On the Galois structure of algebraic integers andS-units,Invent. Math.74 (1983), 321-349. · Zbl 0564.12016 |
| [10] | C.W. Curtis and I. Reiner,Methods of Representation Theory with Applications to Finite Groups and Orders, Vol. 1, Wiley, New York, 1981. · Zbl 0469.20001 |
| [11] | S. Dasgupta, H. Darmon and R. Pollack, Hilbert modular forms and the Gross-Stark conjecture,Ann. of Math.174 (2011), 439-484. · Zbl 1250.11099 |
| [12] | S. Dasgupta, M. Kakde and K. Ventullo, On the Gross-Stark conjecture,Ann. of Math.188 (2018), 833-870. · Zbl 1416.11160 |
| [13] | G. Dejou and X.-F. Roblot, A Brumer-Stark conjecture for non-abelian extensions,J. Number Th.142 (2014), 51-88. · Zbl 1295.11125 |
| [14] | P. Deligne and K.A. Ribet, Values of abelianL-functions at negative integers over totally real fields,Invent. Math.59 (1980), 227-286. · Zbl 0434.12009 |
| [15] | R. Greenberg, Onp-adic ArtinL-functions,Nagoya Math. J.89 (1983), 77-87. · Zbl 0513.12012 |
| [16] | C. Greither, Determining Fitting ideals of minus class groups via the equivariant Tamagawa number conjecture,Compos. Math.143 (2007), 1399-1426. · Zbl 1135.11059 |
| [17] | C. Greither and C.D. Popescu, An equivariant main conjecture in Iwasawa theory and applications,J. Algebraic Geom.24 (2015), 629-692. · Zbl 1330.11070 |
| [18] | B.H. Gross,p-adicL-series atsD0,J. Fac. Sci. Univ. Tokyo Sect. IA Math.28 (1981), 979-994. · Zbl 0507.12010 |
| [19] | K.W. Gruenberg, J. Ritter and A. Weiss, A local approach to Chinburg’s root number conjecture,Proc. London Math. Soc.79 (1999), 47-80. · Zbl 1041.11075 |
| [20] | H. Johnston and A. Nickel, Noncommutative Fitting invariants and improved annihilation results,J. Lond. Math. Soc.88 (2013), 137-160. · Zbl 1290.16016 |
| [21] | H. Johnston and A. Nickel, On the equivariant Tamagawa number conjecture for Tate motives and unconditional annihilation results,Trans. Amer. Math. Soc.368 (2016), 6539-6574. · Zbl 1339.11093 |
| [22] | H. Johnston and A. Nickel, Hybrid Iwasawa algebras and the equivariant Iwasawa main conjecture,Amer. J. Math.140 (2018), 245-276. · Zbl 1395.11122 |
| [23] | H. Johnston and A. Nickel, On the non-abelian Brumer-Stark conjecture and the equivariant Isawana main conjecture,Math. Z.(to appear). · Zbl 1470.11280 |
| [24] | H. Johnston and A. Nickel, On thep-adic Stark conjecture atsD1and applications,preprint, arXiv:1703.06803. |
| [25] | M. Kakde, The main conjecture of Iwasawa theory for totally real fields,Invent. Math.193 (2013), 539-626. · Zbl 1300.11112 |
| [26] | A. Nickel, Non-commutative Fitting invariants and annihilation of class groups,J. Algebra 323 (2010), 2756-2778 · Zbl 1222.11132 |
| [27] | A. Nickel, On non-abelian Stark-type conjectures,Ann. Inst. Fourier61 (2011), 2577-2608. · Zbl 1246.11176 |
| [28] | A. Nickel, On the equivariant Tamagawa number conjecture in tame CM-extensions,Math. Z. 268 (2011), 1-35. · Zbl 1222.11133 |
| [29] | A. Nickel, Equivariant Iwasawa theory and non-abelian Stark-type conjectures,Proc. Lond. Math. Soc.106 (2013), 1223-1247. · Zbl 1273.11155 |
| [30] | A. Nickel, Integrality of Stickelberger elements and the equivariant Tamagawa number conjecture,J. Reine Angew. Math.(Crelle) 719 (2016), 101-132. · Zbl 1359.11088 |
| [31] | J. Nomura, On non-abelian Brumer and Brumer-Stark conjecture for monomial CMextensions,Int. J. Number Th.10 (2014), 817-848. · Zbl 1296.11149 |
| [32] | J. Nomura, The2-part of the non-abelian Brumer-Stark conjecture for extensions with group D4pand numerical examples of the conjecture. InAlgebraic Number Theory and Related Topics 2012(A. Shiho, T. Ochiai and N. Otsubo, eds.), Res. Inst. Math. Sci. (RIMS), Kyoto, 2014, 33-53. · Zbl 1369.11074 |
| [33] | J. Ritter and A. Weiss, On the “main conjecture” of equivariant Iwasawa theory,J. Amer. Math. Soc.24 (2011), 1015-1050. · Zbl 1228.11165 |
| [34] | C.L. Siegel, Über die Fourierschen Koeffizienten von Modulformen,Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II1970 (1970), 15-56. · Zbl 0225.10031 |
| [35] | M. Spiess, Shintani cocycles and the order of vanishing ofp-adic HeckeL-series atsD0, Math. Ann.359 (2014), 239-265. · Zbl 1307.11125 |
| [36] | L. Stickelberger, Ueber eine Verallgemeinerung der Kreistheilung,Math. Ann.37 (1890), 321-367. · JFM 22.0100.01 |
| [37] | J. Tate,Les conjectures de Stark sur les fonctionsLd’Artin ensD0, Lecture notes edited by D. Bernardi and N. Schappacher, Progress in Mathematics 47, Birkhäuser Boston, Boston, 1984. · Zbl 0545.12009 |
| [38] | M. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
