Numerical evidence toward a 2-adic equivariant “Main Conjecture”. (English) Zbl 1312.11087

The purpose of this paper is to test a nonabelian conjecture predicted by the classical \(2\)-adic Main Conjecture of Iwasawa theory. Let \(K\) be a totally real Galois extension of \({\mathbb Q}\) with Galois group \(G\) dihedral of order \(8\) and such that \(\sqrt{2}\notin K\). The authors study a normalized \(2\)-adic function \(F_1(s)\) and \(\rho_{F,S}= \lim_{s\to 1}(s-1)L_{F,S}(s,1)\), where \(L_{F,S}(s,\chi)\) is a suitable normalized \(S\)-truncated \(2\)-adic \(L\)-function, \(S\) a finite set of primes containing \(2\) and \(\infty\), and \(F\) is the fixed field of \(K\) under \(C\), the cyclic subgroup of order \(4\) of \(G\). They consider the conjecture: \(\frac{1}{2}\rho_{F,S}\in 4{\mathbb Z}_2\) and \(F_1(T)\in 4{\mathbb Z}_2[[T]]\). Let \(D(T):=8T(T+2)F_1(T)\). Then the conjecture may be reformulated as: \(\rho_{F,S}\in 8{\mathbb Z}_2\) and \(D(T)\in 32 {\mathbb Z}_2[[T]]\).
The authors verify the conjecture for \(60\) different fields separated into three subcases of 20 examples each, corresponding to the behavior of \(2\) in the quadratic subfield \(F\): ramified, split or inert. In each of the \(60\) examples they compute \(\rho_{F,S}\) and the first \(30\) coefficients of \(D(T)\) to a precision of at least \(2^8\) and checked that they satisfy the conjecture.


11R23 Iwasawa theory
11R70 \(K\)-theory of global fields
11-04 Software, source code, etc. for problems pertaining to number theory


11R23; 11R70
Full Text: DOI arXiv Euclid HAL


[1] Besser [Besser et al. 09] A., Pure Appl. Math. Q. 5 pp 375– (2009)
[2] Cassou-Noguès [Cassou-Noguès 79] P., Invent. Math. 51 pp 29– (1979) · Zbl 0408.12015 · doi:10.1007/BF01389911
[3] Colmez [Colmez 88] P., Invent. Math. 91 pp 371– (1988) · Zbl 0651.12010 · doi:10.1007/BF01389373
[4] Deligne [Deligne and Ribet 80] P., Invent. Math. 59 pp 227– (1980) · Zbl 0434.12009 · doi:10.1007/BF01453237
[5] Ferrero [Ferrero and Washington 79] B., Annals of Math. 109 pp 377– (1979) · Zbl 0443.12001 · doi:10.2307/1971116
[6] Ribet [Ribet 79] K. A., Société Mathém. de France, Astérisque 61 pp 177– (1979)
[7] Ritter [Ritter and Weiss 04] J., Indag. Mathem. N.S. 15 pp 549– (2004) · Zbl 1142.11369 · doi:10.1016/S0019-3577(04)80018-1
[8] Ritter [Ritter and Weiss 06] J., Math. Ann. 336 pp 27– (2006) · Zbl 1154.11038 · doi:10.1007/s00208-006-0773-4
[9] Ritter [Ritter and Weiss 08] J., Documenta Math. 13 pp 117– (2008)
[10] Wiles [Wiles 90] A., Annals of Math. 131 pp 493– (1990) · Zbl 0719.11071 · doi:10.2307/1971468
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.