Let \(F\) be a number field and let \(G_0\) be a non-split inner form of \(\text{GL}(n)/F\) split at a rational prime \(p\) and \(\infty\). Let \({\mathfrak h}^{n,\text{ord}}\) denote the universal \(p\)-adic Hecke algebra attached to \(G= \text{Res}_{F/\mathbb Q} G_0\). The author makes the following conjecture predicting the Krull dimension of \({\mathfrak h}^{n,\text{ord}}\): \[ \dim ({\mathfrak h}^{n,\text{ord}} \otimes{}_{\mathbb Z_p} \mathbb Q_p)\leq \begin{cases} m[F:\mathbb{Q}]+1 &\text{if }n=2m,\\ m[F:\mathbb Q]+ r_2+1 &\text{if }n= 2m+1, \end{cases} \] where \(r_2\) is the number of complex places of \(F\).
The case \(n=1\) reduces to the Leopoldt conjecture for \(p\) and \(F\).
The author proves the conjecture for some inner forms of \(\text{GL}(3)/ \mathbb Q\) and \(\text{GL}(4)/ \mathbb Q\) (section 7.1). The conjecture is checked to be compatible with some of Langlands functorialities (tensor products, base-change, automorphic induction) (section 7.2). If the deformation ring for \(\text{GL}(n)\) is isomorphic to the Hecke algebra, the conjecture is a special case of Tilouine’s conjecture predicting the Krull dimension of the (nearly \(p\)-ordinary) universal deformation ring deforming a fixed \(p\)-adic Galois representation with values in a smooth reductive group over \(\mathbb Z_p\).