Special Year Workshop on p-adic Arithmetic Geometry

Abstract: For a reductive group $G$, its $B_{d}R^{+}$-affine Grassmannian is defined as the étale (equivalently, v-) sheafification of the presheaf quotient $LG/L^{+}G$ of the $B_{d}R$-loop group $LG$ by the $B_{d}R^{+}$-loop subgroup $L^{+}G$. We combine algebraization and approximation techniques with known cases of the Grothendieck–Serre conjecture to show that the analytic topology suffices for this sheafification, more precisely, that the $B_{d}R^{+}$-affine Grassmannian agrees with the analytic sheafification of the presheaf quotient $LG/L^{+}G$. The talk is based on joint work with Alex Youcis.