Let \(K\) be a non-archimedean local field and \(F\) be a function field in one variable over \(K\) with \(\Omega_F\) being the set of all discrete valuations of \(F\) and with \(F_\nu\) denoting the completion of \(F\) at \(\nu\in\Omega_F\). Let \(G\) be a semisimple simply connected linear algebraic group over \(F\). One says that the Hasse principle holds for principal homogeneous spaces \(X\) under \(G\) over \(F\) with respect to \(\Omega_F\) if \(X(F_\nu)\ne\emptyset\) for all \(\nu\in \Omega_F\) implies \(X(F)\ne\emptyset\). This is known to hold for various types of classical groups in the case of \(K\) being a \(p\)-adic field due to work by Y. Hu [J. Ramanujan Math. Soc. 29, No. 2, 155–199 (2014; Zbl 1320.11033)] and R. Preeti [J. Algebra 385, 294–313 (2013; Zbl 1292.11056)]. It is also known for \(G=\mathrm{SL}_1(D)\) for a central simple \(F\)-algebra \(D\) of square free index, as follows from the injectivity of the Rost invariant due to A. S. Merkurjev and A. A. Suslin [Math. USSR, Izv. 36, No. 2, 349–367 (1991); translation from Izv. Akad. Nauk SSSR Ser. Mat. 54, No. 2, 339–356 (1990; Zbl 0711.19003)] and Kato’s Hasse principle for \(H^3(F,\mu_n^{\otimes 2})\) [K. Kato, J. Reine Angew. Math. 366, 142–183 (1986; Zbl 0576.12012)].
In the present paper, the authors establish the Hasse principle for \(G=\mathrm{SL}_1(D)\) for any central simple algebra \(D\) over \(F\) of period coprime to \(p\). More precisely, their first main theorem states that if \(K\) is a local field and \(F\) is a function field of transcendence degree one over \(K\), and if \(D\) is a central simple algebra over \(F\) of period coprime to the residue characteristic of \(K\), then \(\lambda\in F^*\) is a reduced norm from \(D\) whenever \([D]\cdot (\lambda)=0\in H^3(F,\mu_n^{\otimes 2})\). By invoking Kato’s result, they then conclude that if \(\lambda\in F^*\) is a reduced norm from \(D\otimes F_\nu\) for all \(\nu\in \Omega_F\), then \(\lambda\) is a reduced norm from \(D\). They point out that it in fact suffices to restrict to so-called divisorial discrete valuations. The authors also prove an analogous local version of their first main theorem, showing the same statement but now with \(F\) being a complete discretely valued field whose residue field \(\kappa\) is a local or global field and where the period of \(D\) is coprime to the characteristic of \(\kappa\). In the case where \(\kappa\) is a global field, they additionally assume the period of \(D\) to be odd or \(\kappa\) to have no real places.
The proofs make use of the patching techniques developed by D. Harbater, J. Hartmann and D. Krashen [Invent. Math. 178, No. 2, 231–263 (2009; Zbl 1259.12003); Comment. Math. Helv. 89, No. 1, 215–253 (2014; Zbl 1332.11046)], as well as some results from local and global class field theory.