Adelic Loop Groups and Perfectoid Analogies: Factorization and Holomorphic Bundles on the Adelic Projective Line
arXiv:2607.10447
The paper builds a theory of adelic loop groups on a universal solenoid, introduces an adelic projective line with holomorphic vector bundles, and proves factorization and splitting theorems that parallel classical results, while also connecting these ideas to perfectoid geometry.
Abstract
We develop a theory of adelic loop groups on the universal one-dimensional solenoid \(S^1_{\mathbb Q}=(\mathbb R\times\widehat{\mathbb Z})/\mathbb Z_{\mathrm{diag}}\), the compact abelian group whose Pontryagin dual is \(\mathbb Q\) rather than \(\mathbb Z\). We introduce the adelic projective line \(\mathbb{CP}^1_{\mathbb Q}\), its ring of Laurent--Puiseux series, and holomorphic vector bundles defined by solenoidal clutching data. We prove that its Picard group is naturally isomorphic to the additive group \(\mathbb Q\). The paper establishes a scalar Wiener--Birkhoff factorization theorem, a matrix Wiener lemma, exact factorization for ordered triangular and small-norm cocycles, a density theorem for factorable matrix loops in the Wiener algebra \(\mathfrak W_{\mathbb Q}\), and a Birkhoff--Grothendieck splitting theorem in the pro-algebraic category. These results lead to the Solenoidal Birkhoff--Grothendieck conjecture, asserting that every \(g\in \mathrm{GL}*n(\mathfrak W*{\mathbb Q})\) admits a factorization \(g=h_-^{-1}\operatorname{diag}(Ï_{q_1},\ldots,Ï_{q_n})h_+\), where \(h_\pm\in\mathrm{GL}*n(\mathfrak W^\pm*{\mathbb Q})\) and \(q_i\in\mathbb Q\). We also develop the Kahler, Grassmannian, and Morse--Bott geometry of adelic loop groups in the spirit of Pressley--Segal. Finally, we compare the theory with perfectoid geometry. The Fargues--Fontaine curve provides a non-archimedean structural counterpart of \(\mathbb{CP}^1_{\mathbb Q}\) at the level of rational slope data, Kedlaya's slope theory supplies a (p)-adic analogue of Wiener--Birkhoff factorization, and the Fargues--Fontaine classification provides a proved perfectoid model for the matrix splitting problem formulated here. This comparison yields a Harder--Narasimhan reformulation of the Solenoidal Birkhoff--Grothendieck conjecture.