On the Determinantal Approach to the Tamagawa Number Conjecture

Abstract

Abstract We give a survey of Fontaine and Perrin-Riou's formulation of the Tamagawa number conjecture on special values of the L -functions of motives in terms of determinants and Galois cohomology. Following Fontaine's Bourbaki talk, we show its equivalence with the original formulation of Bloch–Kato. As an illustration, we sketch a proof for the Dedekind zeta function of an abelian number field. The conjecture of Bloch and Kato [BK90] on the special values of the L -functions of motives was originally expressed – in analogy with the theory of semi-simple algebraic groups – in terms of Haar measures and Tamagawa numbers. Hence its usual other name, the Tamagawa number conjecture (TNC for short), to which we shall stick in these notes, in order to avoid confusion with another Bloch–Kato conjecture (on K -theory and Galois cohomology; see [Ko15] in this volume). Later on, Fontaine and Perrin-Riou [FPR94] proposed another formulation in terms of determinants of perfect complexes and Galois cohomology. Although the arithmetic becomes less apparent in the new formalism, it allows more flexibility and generality, as illustrated for instance by the subsequent development of the equivariant version of the conjecture (ETNC for short), which ‘provides a coherent overview and refinement of many existing “equivariant” conjectures, including for example the refined Birch–Swinnerton-Dyer conjecture for CM elliptic curves formulated by Gross, the conjectural congruences of Dirichlet L -functions formulated by Gross and Tate, the conjectures formulated by Chinburg et al . in the area of Galois module theory’ (see [BG03, Introduction, p.303]). As for the TNC proper, Fontaine and Perrin-Riou note that ‘the complicated formulas bringing in Tamagawa numbers, orders of Shafarevich groups, are only the consequence of the explicit calculation of an “intrinsic” formula making use of certain Euler–Poincare characteristics’ [FPR94, p. 600]. Being among the arithmeticians who regret the occultation of these ‘complicated formulas’ in the style of the analytic class number formula, we were too happy to accept the proposal of the organizers of the Pune workshop to write these notes on the comparison and (at least when the field of coefficients is ℚ) the equivalence between the two formulations, that of Bloch–Kato and that of Fontaine–Perrin-Riou.