Height zeta functions of equivariant compactifications of semi-direct products of algebraic groups (Q2918494)

The authors explain the basic principles of spectral theory on adelic groups and of the method of height zeta-functions, as applied to counting rational points of equivariant compactifications of linear algebraic groups, in particular, of semi-direct products. They show that the main term appearing in the spectral analysis, namely, the term corresponding to one-dimensional representations, matches presisely the predictions of the Batyrev-Manin conjecture. Let \(G_1\) be the semi-direct product of the additive group \(G_a\) and the multiplicative group \(G_m\), with the group law defined as follows: NEWLINE\[NEWLINE(x,u)\cdot(y,v)= (x+ uy,uv),NEWLINE\]NEWLINE and let \(X\) be a smooth projective equivariant compactification of \(G_1\), under the right action, defined over \(\mathbb{Q}\). Under somewhat technical geometric conditions on \(X\), the authors prove that variety \(X\) satisfies the Batyrev-Manin conjecture. Their techniques rely heavily on the previous works of the second author and his collaborators.NEWLINENEWLINEFor the entire collection see [Zbl 1242.11004].