\(p\)-adic automorphic functions for the unitary group in three variables (Q2711611)

The general pattern for the construction of interesting rigid spaces over a non-archimedean local field \(K\) goes as follows. An action of a linear algebraic group \(G\) over \(K\) on some projective space \(Y\) over \(K\) is given. One considers a suitable rigid open subspace \(Y^{s}\) of \(Y\) and a discrete subgroup \(\Gamma \subset G(K)\) which is co-compact or has finite co-volume. The resulting rigid space \(Y^s/\Gamma\) can be of interest for algebraic geometry. Examples: the Tate elliptic curve, Abelian varieties with multiplicative reduction, Mumford curves, Drinfeld's moduli spaces for elliptic modules etc. In many cases the space \(Y^s/\Gamma\) will not be proper and/or not be algebraizable. In extreme cases, the only meromorphic functions on \(Y^s/\Gamma\) are the constant functions. NEWLINENEWLINENEWLINEIn this paper one considers an extension of local fields \(K\subset L\) of degree 2 (not equal to the residue characteristic of \(K\)), the algebraic group \(\text{SU}_3\) and a discrete co-compact subgroup \(\Gamma\) of \(\text{SU}_3(L)\). The group \(\text{SU}_3(L)\) acts on \(\mathbb{P}^2_L\) and the rigid open subspace \(Y^s\) consists of the points which are stable for the action of every \(K\)-split maximal torus of \(\text{ SU}_3(L)\). The combinatorial structure of the rigid space \(Y^r\) is made explicit by a map \(I\) from \(Y^r\) to the Bruhat-Tits building of \(\text{SU}_3(L)\), which is a tree. This map assignes to every point \(x\) a bounded convex subset \(I(x)\) of the building. This interval of semistability describes the semistability of \(x\) extended as section over the ring of integers of \(L\). The map \(I\) is used to construct non-trival meromorphic functions on \(Y^r/\Gamma\). One construction produces \(\Gamma\)-invariant infinite products on \(Y^r\), somewhat analogous to theta-functions for a Schottky group. The other construction produces \(\Gamma\)-invariant infinite ``Poincaré ''sums on \(Y^r\). It would be interesting to know whether the field of meromorphic functions on \(Y^r/\Gamma\) is finitely generated over \(L\) and of transcendence degree 2.