On the Euler-Poincaré characteristics of finite dimensional \(p\)-adic Galois representations.

*(English)*Zbl 1143.11021Let \(p\) be a prime number, and let \(V\) be a finite-dimensional vector space over the field \(\mathbb{Q}_p\) of \(p\)-adic numbers. For a compact subgroup \(G_V\) of the general linear group \(\text{GL}(V)\) such that \(G_V\) is a \(p\)-adic Lie group, denote by \(H^i(G_V, V)\) the cohomology groups of \(G_V\) acting on \(V\), that is, the groups defined by continuous \(i\)-cochains with respect to \(p\)-adic topology of \(V\). The representation \(V\) of \(G_V\) is said to have vanishing \(G_V\)-cohomology, iff \(H^i(G_V , V)= 0\) for all \(i\geq 0\).

The first interesting example of such a representation arising in arithmetic geometry was studied by J.-P. Serre back in 1971 [Izv. Akad. Nauk SSSR, Ser. Mat. 35, 731–737 (1971; Zbl 0222.14025)].

In this context, one of the aims of the present paper is to show how a broad class of new examples of this type can be obtained from the étale cohomology of smooth proper algebraic varieties defined over a finite extension of the field \(\mathbb{Q}_p\) and having potential good reduction. To this end, the authors first generalize some basic properties of Lie algebra and Lie group cohomology, which are similar in spirit to those already used in their foregoing paper “Euler-Poincaré characteristics of Abelian varieties” [C. R. Acad. Sci., Paris, Sér. I, Math. 329, No. 4, 309–313 (1999; Zbl 0967.14029)]. In fact, the general arguments provided here essentially underly the proofs of the authors’ main results derived in the sequel. Among those are the following fundamental new theorems:

(1) Let \(F\) be a finite extension field of \(\mathbb{Q}_p\) and let \(Y\) be a smooth proper variety defined over \(F\) with potential good reduction. For any two integers \(i\geq 0\) and \(j\neq{i\over 2}\), put \[ V:= H^i_{\text{ét}}(Y_{\mathbb{Q}_p}, \mathbb{Q}_p(j))\quad\text{and}\quad G_V:= \rho(G_F), \] where \(\rho: G_F\to\text{GL}(H^i_{\text{ét}}(Y_{\mathbb{Q}_p}, \mathbb{Q}_p(j))\) stands for the representation giving the canonical action of \(G_F\) on these vector spaces. Then \(G_V\) has vanishing \(G_V\)-cohomology. Moreover, if \(V'\) is any Galois subquotient of \(V\) then \(V'\) has both vanishing \(G_V\)-cohomology and vanishing \(G_{V'}\)-cohomology.

(2) If \(V'\) is a finite-dimensional continuous representation of \(G_V\) over \(\mathbb{Q}_p\) and if \(G_V\) has no element of order \(p\), then there is a well-defined cohomological Euler-Poincaré characteristic \(\chi_f(G_V, V')\) for these data, which turns out to be equal to 1 in certain arithmetic-geometric situations.

(3) If \(F\) and \(Y\) are given as in (1), and if \(H_V:= \rho(G_{F_\infty})\), where \(G_{F_\infty}\) denotes the Galois group of \(\mathbb{Q}_p\) over the cyclotomic \(\mathbb{Z}_p\)-extension \(F_\infty\) of \(F\), then \(V\) has vanishing \(H_V\)-cohomology for each odd integer \(i\geq 1\). Moreover, if \(V'\) is any Galois subquotient representation of \(V\) then \(V'\) has vanishing \(H_V\)- cohomology.

Further new results in this direction, mainly with regard to vanishing \(G_V\)-cohomology or to the triviality of Euler-Poincaré characteristics for special types of \(p\)-adic Galois representations, are derived in the course of the paper, together with various applications to a large class of motivic Galois representations, on the one hand, and to elliptic curves \(A\) over a finite extension \(F\) over \(\mathbb{Q}_p\) on the other.

Apart from the basic toolkit of Lie algebra and Lie group cohomology, the authors use the theory of semistable representations due to J.-M. Fontaine [\(p\)-adic semi-stable representations. (Exposé III: Représentations \(p\)-adiques semi-stables.), Astérisque 223, 113–184 (1994; Zbl 0865.14009)] in a crucial way.

As for the applications to elliptic curves, the authors’ approach is essentially motivated by the parallel work of J. Coates and S. Howson titled “Euler characteristics and elliptic curves. II” [J. Math. Soc. Japan 53, No. 1, 175–235 (2001; Zbl 1046.11079)]. Finally, a different approach to computing Euler characteristics for \(p\)-adic Lie groups has been proposed by B. Totaro [Publ. Math., Inst. Hautes Étud. Sci. 90, 169–225 (1999; Zbl 0971.22011)] shortly before and in the same journal. However, as the authors point out, it seems to be quite difficult to compare their results to those of B. Totaro’s.

The first interesting example of such a representation arising in arithmetic geometry was studied by J.-P. Serre back in 1971 [Izv. Akad. Nauk SSSR, Ser. Mat. 35, 731–737 (1971; Zbl 0222.14025)].

In this context, one of the aims of the present paper is to show how a broad class of new examples of this type can be obtained from the étale cohomology of smooth proper algebraic varieties defined over a finite extension of the field \(\mathbb{Q}_p\) and having potential good reduction. To this end, the authors first generalize some basic properties of Lie algebra and Lie group cohomology, which are similar in spirit to those already used in their foregoing paper “Euler-Poincaré characteristics of Abelian varieties” [C. R. Acad. Sci., Paris, Sér. I, Math. 329, No. 4, 309–313 (1999; Zbl 0967.14029)]. In fact, the general arguments provided here essentially underly the proofs of the authors’ main results derived in the sequel. Among those are the following fundamental new theorems:

(1) Let \(F\) be a finite extension field of \(\mathbb{Q}_p\) and let \(Y\) be a smooth proper variety defined over \(F\) with potential good reduction. For any two integers \(i\geq 0\) and \(j\neq{i\over 2}\), put \[ V:= H^i_{\text{ét}}(Y_{\mathbb{Q}_p}, \mathbb{Q}_p(j))\quad\text{and}\quad G_V:= \rho(G_F), \] where \(\rho: G_F\to\text{GL}(H^i_{\text{ét}}(Y_{\mathbb{Q}_p}, \mathbb{Q}_p(j))\) stands for the representation giving the canonical action of \(G_F\) on these vector spaces. Then \(G_V\) has vanishing \(G_V\)-cohomology. Moreover, if \(V'\) is any Galois subquotient of \(V\) then \(V'\) has both vanishing \(G_V\)-cohomology and vanishing \(G_{V'}\)-cohomology.

(2) If \(V'\) is a finite-dimensional continuous representation of \(G_V\) over \(\mathbb{Q}_p\) and if \(G_V\) has no element of order \(p\), then there is a well-defined cohomological Euler-Poincaré characteristic \(\chi_f(G_V, V')\) for these data, which turns out to be equal to 1 in certain arithmetic-geometric situations.

(3) If \(F\) and \(Y\) are given as in (1), and if \(H_V:= \rho(G_{F_\infty})\), where \(G_{F_\infty}\) denotes the Galois group of \(\mathbb{Q}_p\) over the cyclotomic \(\mathbb{Z}_p\)-extension \(F_\infty\) of \(F\), then \(V\) has vanishing \(H_V\)-cohomology for each odd integer \(i\geq 1\). Moreover, if \(V'\) is any Galois subquotient representation of \(V\) then \(V'\) has vanishing \(H_V\)- cohomology.

Further new results in this direction, mainly with regard to vanishing \(G_V\)-cohomology or to the triviality of Euler-Poincaré characteristics for special types of \(p\)-adic Galois representations, are derived in the course of the paper, together with various applications to a large class of motivic Galois representations, on the one hand, and to elliptic curves \(A\) over a finite extension \(F\) over \(\mathbb{Q}_p\) on the other.

Apart from the basic toolkit of Lie algebra and Lie group cohomology, the authors use the theory of semistable representations due to J.-M. Fontaine [\(p\)-adic semi-stable representations. (Exposé III: Représentations \(p\)-adiques semi-stables.), Astérisque 223, 113–184 (1994; Zbl 0865.14009)] in a crucial way.

As for the applications to elliptic curves, the authors’ approach is essentially motivated by the parallel work of J. Coates and S. Howson titled “Euler characteristics and elliptic curves. II” [J. Math. Soc. Japan 53, No. 1, 175–235 (2001; Zbl 1046.11079)]. Finally, a different approach to computing Euler characteristics for \(p\)-adic Lie groups has been proposed by B. Totaro [Publ. Math., Inst. Hautes Étud. Sci. 90, 169–225 (1999; Zbl 0971.22011)] shortly before and in the same journal. However, as the authors point out, it seems to be quite difficult to compare their results to those of B. Totaro’s.

Reviewer: Werner Kleinert (Berlin)

##### MSC:

11F80 | Galois representations |

11F85 | \(p\)-adic theory, local fields |

14G20 | Local ground fields in algebraic geometry |

14F20 | Étale and other Grothendieck topologies and (co)homologies |

14F42 | Motivic cohomology; motivic homotopy theory |

11G07 | Elliptic curves over local fields |