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**Essential \(p\)-dimension of algebraic groups whose connected component is a torus.**
*(English)*
Zbl 1288.20061

This paper is a continuation of previous work by the same authors [J. Reine Angew. Math. 677, 1-13 (2013; Zbl 1354.14071)]. Let \(p\) be a prime and let \(k\) be a field of characteristic different from \(p\). Let \(G\) be an algebraic group over \(k\) whose component \(G^0\) containing the identity is an algebraic torus. Using various reductions, the authors are able to work with the assumption that \(k\) contains a primitive \(p\)-th root of unity, the quotient \(G/G^0\) is a finite \(p\)-group, and that \(T\) is split and \(G/G^0\) becomes constant over a field extension of \(k\) with degree a power of \(p\).

The main result of the paper (Theorem 1.2) gives upper and lower bounds for the essential \(p\)-dimension of \(G\). The lower and upper bounds are given in terms of the dimensions of various representations of \(G\). The proof for the upper bound is quite simple, and uses the fact (Proposition 3.1) that if there exists an isogeny between algebraic groups of degree prime to \(p\) then the two groups have equal essential \(p\)-dimension. The proof for the lower bound uses a certain central subgroup of \(G\) and a previous result of the authors (reproduced in Theorem 4.1).

The authors call the difference between the upper bound and the lower bound given by Theorem 1.2 the ‘gap’. Theorem 1.3 gives an upper bound on this gap. It is also proven that the gap vanishes for a class of groups called ‘tame’ (Definition 7.3). The last main result of the paper is Theorem 1.4, which is an additivity result for groups of zero gap. In this case, the gap of the product is also zero, and the essential \(p\)-dimension of the product is the sum of the essential \(p\)-dimensions of the two factors. The paper ends with a study of groups (that are extensions of finite \(p\)-groups by tori) having low essential \(p\)-dimension \(r\) (with \(0\leq r\leq r-2\)) and Proposition 10.6 gives equivalent conditions for the essential \(p\)-dimension to be \(r\).

The main result of the paper (Theorem 1.2) gives upper and lower bounds for the essential \(p\)-dimension of \(G\). The lower and upper bounds are given in terms of the dimensions of various representations of \(G\). The proof for the upper bound is quite simple, and uses the fact (Proposition 3.1) that if there exists an isogeny between algebraic groups of degree prime to \(p\) then the two groups have equal essential \(p\)-dimension. The proof for the lower bound uses a certain central subgroup of \(G\) and a previous result of the authors (reproduced in Theorem 4.1).

The authors call the difference between the upper bound and the lower bound given by Theorem 1.2 the ‘gap’. Theorem 1.3 gives an upper bound on this gap. It is also proven that the gap vanishes for a class of groups called ‘tame’ (Definition 7.3). The last main result of the paper is Theorem 1.4, which is an additivity result for groups of zero gap. In this case, the gap of the product is also zero, and the essential \(p\)-dimension of the product is the sum of the essential \(p\)-dimensions of the two factors. The paper ends with a study of groups (that are extensions of finite \(p\)-groups by tori) having low essential \(p\)-dimension \(r\) (with \(0\leq r\leq r-2\)) and Proposition 10.6 gives equivalent conditions for the essential \(p\)-dimension to be \(r\).

Reviewer: Emre Coskun (Ankara)

### MSC:

20G15 | Linear algebraic groups over arbitrary fields |

11E72 | Galois cohomology of linear algebraic groups |

20G05 | Representation theory for linear algebraic groups |