## SELMER GROUPS AND CHOW GROUPS OF SELF-PRODUCTS OF ALGEBRAIC VARIETIES

### BibTeX

@MISC{Weston_selmergroups,

author = {Tom Weston},

title = {SELMER GROUPS AND CHOW GROUPS OF SELF-PRODUCTS OF ALGEBRAIC VARIETIES},

year = {}

}

### OpenURL

### Abstract

Abstract. Let X be a proper flat scheme over the ring of integers of a global field. We show that the Tate conjecture and the finiteness of the Chow group of vertical cycles on self-products of X implies the vanishing of the dual Selmer group of certain twists of tensor powers of representations occurring in the étale cohomology of X. 1.

### Citations

373 |
Representation theory, A first course
- Fulton, Harris
- 1991
(Show Context)
Citation Context ...χ with ε the cyclotomic character and χ of finite order. It follows that V ⊗r (d) ⊗ χ is the realization of the Galois representation (π ⊗r ⊗ µ −r/κ ) ◦ ρ. By the theory of Schur representations (see =-=[5]-=-), the representation π ⊗r ⊗ µ −r/κ is semisimple; as ρ(GF ) is open (and thus Zariski dense) in GV (K), the Galois representation V ⊗r (d) ⊗ χ is then semisimple as well. Let λ be an irreducible cons... |

284 |
La conjecture de weil
- Deligne
- 1974
(Show Context)
Citation Context ... last condition as vacuous.) Let W be an irreducible Qℓ[GF ]-quotient of Hm ét ( ¯ X, Qℓ) for some m (with ¯ X the base change of X to a separable closure of F ). By work of Deligne and Faltings (see =-=[1]-=- and [2]) W and V := HomQℓ (W, Qℓ) satisfy the hypotheses of Section 3.1 with weights m and −m, respectively. Fix a maximal subfield K ↩→ EndQℓ[GF ] V . From now on we regard V as an absolutely irredu... |

76 | A First Course in Noncommutative Rings - Lam - 2001 |

63 |
Crystalline Cohomology and p-adic Galois Representations
- Faltings
- 1990
(Show Context)
Citation Context ...ndition as vacuous.) Let W be an irreducible Qℓ[GF ]-quotient of Hm ét ( ¯ X, Qℓ) for some m (with ¯ X the base change of X to a separable closure of F ). By work of Deligne and Faltings (see [1] and =-=[2]-=-) W and V := HomQℓ (W, Qℓ) satisfy the hypotheses of Section 3.1 with weights m and −m, respectively. Fix a maximal subfield K ↩→ EndQℓ[GF ] V . From now on we regard V as an absolutely irreducible K[... |

36 |
A finiteness theorem for the symmetric square of an elliptic curve
- Flach
- 1992
(Show Context)
Citation Context ...e show that if enough vertical cycles are torsion, then one can control Selmer groups. In many ways this theorem can be regarded as a massive generalization of part of Flach’s finiteness theorem (see =-=[3]-=-) for Selmer groups of symmetric squares of elliptic curves. In our notation, Flach considers the case F = Q, X an elliptic curve, V its Tate space, and r = 2. He uses a modular unit construction to s... |

34 |
Propriétés conjecturales des groupes de Galois motiviques et des représentations l-adiques,” Motives
- Serre
- 1991
(Show Context)
Citation Context ...e standard representation of GLN , GSp N or GON over ¯K. (In particular, GV ×K ¯ K is then isomorphic to one of these groups.) It is conjectured that the openness condition holds if ρ is motivic; see =-=[9]-=- for details. 3.3. Locally isotropic Galois representations. We say that the Galois representation V is locally isotropic if there is an open subset U ⊆ GF such that each γ ∈ U fixes some non-zero vec... |

16 |
Valeurs spéciales des fonctions L des motifs, Astérisque 206
- Fontaine
- 1992
(Show Context)
Citation Context ..., V ) → ∏ v /∈P v /∈P H 1 ) s (Fv, V ) ; we simply write S(F, V ) for Sφ(F, V ). For motivic V it is conjectured that S(F, V ) is related to the order of the L-function of the dual of V at s = 1; see =-=[4]-=-. Remark 3.2. One can analogusly define S(F, V/T ) for a Galois stable lattice T in V , and one sees easily that S(F, V/T ) is finite if and only if S(F, V ) = 0.6 TOM WESTON Set W = HomK(V, K(1)). W... |

4 |
Finiteness of torsion in the codimension-two Chow group: an axiomatic approach
- Langer
- 1988
(Show Context)
Citation Context ... dimension of a Selmer group and the order of a vanishing of an L-function). See [10, Section 4.1] for a discussion. In addition, the results of this paper are in some sense complementary to those of =-=[7]-=- and [8], where results on Selmer groups are used to control torsion cycles on surfaces. Here, by contrast, we show that if enough vertical cycles are torsion, then one can control Selmer groups. In m... |

2 | Zero-cycles on Hilbert-Blumenthal surfaces
- Langer
(Show Context)
Citation Context ...on of a Selmer group and the order of a vanishing of an L-function). See [10, Section 4.1] for a discussion. In addition, the results of this paper are in some sense complementary to those of [7] and =-=[8]-=-, where results on Selmer groups are used to control torsion cycles on surfaces. Here, by contrast, we show that if enough vertical cycles are torsion, then one can control Selmer groups. In many ways... |

1 |
Geometric Euler systems for locally isotropic motives
- Weston
- 2001
(Show Context)
Citation Context ...o produce a geometric Euler system? Second, what properties of the representation V ⊗2 (−1) allow one to use the geometric Euler system to control the Selmer group? The latter question is answered in =-=[10]-=-: a geometric Euler system can be used to control the Selmer group of a locally isotropic Galois representation. (An irreducible Galois representation is said to be locally isotropic if there is an op... |