Iwasawa theory and motivic \(L\)-functions
This paper is a continuation of the survey article by the author in [Contemp. Math. 358, 79--125 (2004; Zbl 1070.11025)], on the Equivariant Tamagawa Number Conjecture (ETNC). Without pretending to give a precise statement of the ETNC, let us just say that a pure motif over \({\mathbb Q}\) on which a semi-simple \({\mathbb Q}\)-algebra \(A\) acts, gives rise to a ``motivic structure'' consisting of the realizations and the motivic cohomology groups of \(M\) together with comparison isomorphisms and exact sequences relating these groups. There is then a whole corpus of conjectures (no less than eight are listed in this paper) which describe, on the one hand, the motivic cohomology with \({\mathbb R}\)-coefficients and \({\mathbb Q}_\ell\)-coefficients, on the other hand, the behaviour at \(s = 0\) of a hypothetical complex \(L\)-function \(L({}_\wedge M, s).\) Of particular importance are a rationality conjecture and an integrality conjecture for the leading term \(L^\ast({}_\wedge M).\) Note that all these conjectures have been completely (resp. essentially) proved for Tate motives (resp. motives associated to CM elliptic curves) over abelian extensions of \({\mathbb Q}\) (resp. of an imaginary quadratic field) thanks to the efforts of many people, culminating in \textit{D. Burns} and \textit{C. Greither}'s [Invent. Math. 153, No. 2, 303--359 (2003; Zbl 1142.11076), resp. \textit{W. Bley}, Doc. Math., J. DMV 11, 73--118 (2006; Zbl 1178.11070)]. The ``modern'' point of view due to Kato, Fontaine--Perrin-Riou, Burns--Flach et al., presents both the theory of complex \(L\)-values as well as of \(p\)-adic \(L\)-functions as the construction of ``trivializations'' of determinant line bundles. The integrality conjecture pertains mainly to Iwasawa theory via the Main Conjecture, but to address it, it is not sufficient to know the rationality conjecture as such (i.e. up to an unspecified factor in \(\wedge^\ast),\) a more precise description of \(L^\ast({}_\wedge M)\) is needed. Any such description is called by the author a ``limit formula'' (the terminology is purposely kept vague). Known examples are the classical analytic formulas for Dirichlet \(L\)-functions , or the Kronecker limit formula for \(L\)-functions associated to an imaginary quadratic field (resp. for CM elliptic curves at the central point). In the main part of this paper, the author applies the above strategy (trivializations of determinants and limit formulas) to the Iwasawa theory of quadratic fields, more precisely to the Main Conjecture for imaginary quadratic fields and Wiles' explicit reciprocity law (in Kato's formulation in terms of the dual exponential map). A short concluding paragraph explains how the conjectural picture could be extended to the non commutative case.
scientific article; zbMATH DE number 5543150