We construct an fpqc gerbe $\mathcal {E}_{\dot {V}}$ over a global function field F such that for a connected reductive group G over F with finite central subgroup Z, the set of $G_{\mathcal {E}_{\dot {V}}}$-torsors contains a subset $H^{1}(\mathcal {E}_{\dot {V}}, Z \to G)$ which allows one to define a global notion of (Z-)rigid inner forms. There is a localization map $H^{1}(\mathcal {E}_{\dot {V}}, Z \to G) \to H^{1}(\mathcal {E}_{v}, Z \to G)$, where the latter parametrizes local rigid inner forms (cf. [8, 6]) which allows us to organize local rigid inner forms across all places v into coherent families. Doing so enables a construction of (conjectural) global L-packets and a conjectural formula for the multiplicity of an automorphic representation $\pi $ in the discrete spectrum of G in terms of these L-packets. We also show that, for a connected reductive group G over a global function field F, the adelic transfer factor $\Delta _{\mathbb {A}}$ for the ring of adeles $\mathbb {A}$ of F serving an endoscopic datum for G decomposes as the product of the normalized local transfer factors from [6].

Let p be a fixed prime number, and let F be a global function field with characteristic not equal to p. In this article, we shall study the variation properties of the Sylow p-subgroups of the even K-groups in a p-adic Lie extension of F. When the p-adic Lie extension is assumed to contain the cyclotomic $\mathbb {Z}_p$-extension of F, we obtain growth estimate of these groups. We also establish a duality between the direct limit and inverse limit of the even K-groups.

For every affine variety over a global function field, we show that the set of its points with coordinates in an arbitrary rank-one multiplicative subgroup of this function field satisfies the required property of weak approximation for finite sets of places of this function field avoiding arbitrarily given finitely many places.

Let $k = {\Bbb F}_q(t)$ be the rational function field with finite constant field and characteristic $p \geq 3$, and let K/k be a finite separable extension. For a fixed place v of k and an elliptic curve E/K which has ordinary reduction at all places of K extending v, we consider a canonical height pairing $\langle \phantom {x},\phantom {x}\! \rangle _v \colon E(K^{\rm {sep}}) \times E(K^{\rm {sep}}) \to {\Bbb C}_{v}^{\times }$ which is symmetric, bilinear and Galois equivariant. The pairing $\langle \phantom {x},\phantom {x}\! \rangle _\infty$ for the “infinite” place of k is a natural extension of the classical Néron–Tate height. For v finite, the pairing $\langle \phantom {x},\phantom {x}\! \rangle _v$ plays the role of global analytic p-adic heights. We further determine some hypotheses for the nondegeneracy of these pairings.