Iwasawa theory of crystalline representations. II. (Théorie d’Iwasawa des représentations cristallines. II.) (French) Zbl 1157.11041
Let \(K\) be a finite unramified extension of \({\mathbb Q}_p\) (\(p\) odd) and \(V\) a crystalline representation of \(G_K.\) The literature on \(p\)-adic representations contains three important related conjectures, denoted respectively \(C_{EP}(K, V)\) (Fontaine-Perrin-Riou), \(C_{EP} (L/K, V)\) (or Kato’s “local \(\varepsilon\)-conjecture”) and \(C_{Iw}(K_\infty/K, V)\) (or Perrin-Riou’s “\(\delta_{{\mathbb Z}p}(V)\) conjecture”). Without recalling all the definitions, let us describe briefly the content of these conjectures:
Let \(L/K\) be a finite abelian extension, \(G = \text{Gal}(L/K).\) Suppose that the representation \(V\) is potentially semi-stable and take a \(G\)-lattice \(T\) contained in \(V.\) There is a canonical trivialisation of the Euler-Poincaré line \(\delta_{V, L/K} : \Delta_{EP}(L/K,V) \simeq {\mathbb Q}_p [G]_{V, L/K},\) the right hand side being a certain free \({\mathbb Q}_p[G]\)-module of rank 1 which contains a canonical invertible \({\mathbb Z}_p [G]\)-submodule \({\mathbb Z}_p [G]_{V, L/K}.\) The conjecture \(C_{EP} (L/K, V)\) states that \(\delta_{V, L/K}\) sends \(\Delta_{EP} (L/K, T)\) onto \({\mathbb Z}_p [G]_{V, L/K}\) (note that the image of \(\Delta_{EP} (L/K, T)\) inside \(\Delta_{EP} (L/K, V)\) does not depend on the choice of \(T).\) Conjecture \(C_{EP} (K, V)\) is of course a particular case of \(C_{EP}(L/K,V).\) Now consider \(K_\infty = K (\xi_{p^\infty})\) and the “infinite” version \(\Delta_{Iw} (K_\infty/K, V)\) of the Euler-Poincaré line. Perrin-Riou’s reciprocity law gives a canonical isomorphism \[ \delta_{V, K_\infty/K} : \Delta_{Iw} (K_\infty/K, V) \;{\displaystyle\buildrel\sim\over\to} \;{\mathbb Q}_p \displaystyle\bigotimes_{{\mathbb Z}_p} \wedge_{V, K_\infty/K}, \] where \(\wedge_{V, K_\infty/K}\) is a certain \(\wedge\)-free module of rank 1. The conjecture \(C_{Iw} (K_\infty/K, V)\) states that, for a crystalline representation \(V,\) \(\delta_{V, K_\infty/K}\) sends \(\Delta_{Iw} {(K_\infty/K, T)}\) onto \(\wedge_{V, K_\infty/K}\) (this is actually an integrality conjecture concerning Perrin-Riou’s “large exponential map”).
In this paper, the authors prove that \(C_{Iw} (K_\infty/K, V)\) is true by using the theory of \((\varphi, \Gamma)\)-modules. Iwasawa-theoretic techniques then imply that \(C_{EP} (L/K, V)\) is true for any finite extension \(L\) of \(K\) contained in \(K_\infty.\) Beyond the theory of \(p\)-adic representations proper, the interest of these conjectures is that they show that the Tamagawa number conjecture (equivariant or not) on special values of \(L\)-functions is compatible with the functional equation. For particular cases, see e.g. [D. Benois, T. Nguyen Quang Do, Ann. Sci. Éc. Norm. Supér. (4) 35, No. 5, 641–672 (2002; Zbl 1125.11351)], or [D. Burns, M. Flach, Doc. Math., J. DMV Extra Vol., 133–163 (2006; Zbl 1156.11042)].
Part I, cf. D. Benois [Duke Math. J. 104, 211-267 (2000; Zbl 0996.11072)].
Let \(L/K\) be a finite abelian extension, \(G = \text{Gal}(L/K).\) Suppose that the representation \(V\) is potentially semi-stable and take a \(G\)-lattice \(T\) contained in \(V.\) There is a canonical trivialisation of the Euler-Poincaré line \(\delta_{V, L/K} : \Delta_{EP}(L/K,V) \simeq {\mathbb Q}_p [G]_{V, L/K},\) the right hand side being a certain free \({\mathbb Q}_p[G]\)-module of rank 1 which contains a canonical invertible \({\mathbb Z}_p [G]\)-submodule \({\mathbb Z}_p [G]_{V, L/K}.\) The conjecture \(C_{EP} (L/K, V)\) states that \(\delta_{V, L/K}\) sends \(\Delta_{EP} (L/K, T)\) onto \({\mathbb Z}_p [G]_{V, L/K}\) (note that the image of \(\Delta_{EP} (L/K, T)\) inside \(\Delta_{EP} (L/K, V)\) does not depend on the choice of \(T).\) Conjecture \(C_{EP} (K, V)\) is of course a particular case of \(C_{EP}(L/K,V).\) Now consider \(K_\infty = K (\xi_{p^\infty})\) and the “infinite” version \(\Delta_{Iw} (K_\infty/K, V)\) of the Euler-Poincaré line. Perrin-Riou’s reciprocity law gives a canonical isomorphism \[ \delta_{V, K_\infty/K} : \Delta_{Iw} (K_\infty/K, V) \;{\displaystyle\buildrel\sim\over\to} \;{\mathbb Q}_p \displaystyle\bigotimes_{{\mathbb Z}_p} \wedge_{V, K_\infty/K}, \] where \(\wedge_{V, K_\infty/K}\) is a certain \(\wedge\)-free module of rank 1. The conjecture \(C_{Iw} (K_\infty/K, V)\) states that, for a crystalline representation \(V,\) \(\delta_{V, K_\infty/K}\) sends \(\Delta_{Iw} {(K_\infty/K, T)}\) onto \(\wedge_{V, K_\infty/K}\) (this is actually an integrality conjecture concerning Perrin-Riou’s “large exponential map”).
In this paper, the authors prove that \(C_{Iw} (K_\infty/K, V)\) is true by using the theory of \((\varphi, \Gamma)\)-modules. Iwasawa-theoretic techniques then imply that \(C_{EP} (L/K, V)\) is true for any finite extension \(L\) of \(K\) contained in \(K_\infty.\) Beyond the theory of \(p\)-adic representations proper, the interest of these conjectures is that they show that the Tamagawa number conjecture (equivariant or not) on special values of \(L\)-functions is compatible with the functional equation. For particular cases, see e.g. [D. Benois, T. Nguyen Quang Do, Ann. Sci. Éc. Norm. Supér. (4) 35, No. 5, 641–672 (2002; Zbl 1125.11351)], or [D. Burns, M. Flach, Doc. Math., J. DMV Extra Vol., 133–163 (2006; Zbl 1156.11042)].
Part I, cf. D. Benois [Duke Math. J. 104, 211-267 (2000; Zbl 0996.11072)].
Reviewer: Thong Nguyen Quang Do (Besançon)
