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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.

MSC:

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

Keywords:

11R23; 11R70
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