Upper Bounds for the Essential Dimension of E₇

Abstract

Abstract. This paper gives a new upper bound for the essential dimension and the essential 2-dimension of the split simply connected group of type E 7 over a field of characteristic not 2 or 3. In particular, ed(E 7 ) ≤ 29, and ed(E 7 ; 2) ≤ 27.

The slice method for G-torsors

“…In this section E 7 will denote the split simply connected algebraic group of type E 7 . The best published bounds on the essential dimension of this group are 7 ≤ ed(E 7 ) ≤ 29, where the lower bound is valid for fields of characteristic not 2 [CS06], and the upper bound is valid for fields of characteristic not 2 or 3 [Mac13]. We will show that ed(E 7 ) ≤ 11 for all fields of characteristic not 2 or 3.…”

“…26 . Also, Spin 8 is an SGP of E 6 acting on V 56 , by [Mac13,Prop.1.3] together with Lemma 2.13. Therefore, Spin 8 is an SGP of the action of E 7 on V .…”

“…Firstly, the adjoint representation has a 7-dimensional slice which gives a reduction of structure to the normalizer of a maximal torus. Secondly, V 56 has SGP E 6 , so by Theorem 4.1 there is a 1-dimensional slice which gives a reduction of structure to Norm E7 (E 6 ); also the 0-dimensional slice of the versal P(V 56 ) gives a reduction to E 6 µ 4 , which was considered in [Ga09] and [Mac13]. Thirdly, we have V := V 56 ⊕ V 56 , which we consider below.…”