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No abstract

Let X be a smooth, geometrically connected curve over a perfect field k. Given a connected, reductive group G, we prove that central extensions of G by the sheaf K 2 on the big Zariski site of X, studied in Brylinski-Deligne [BD01], are equivalent to factorization line bundles on the Beilinson-Drinfeld affine Grassmannian Gr G . Our result affirms a conjecture of Gaitsgory-Lysenko [GL16] and classifies factorization line bundles on Gr G . the loop group by the work of R. Reich [Re12]. Finally, one obtains (0.2) by taking the trace of Frobenius. 0.2.5. The association of covering groups to Brylinski-Deligne data is thus seen to factor as the following composition of functors: Brylinski-Deligne data ΦG − − → factorization line bundles on Gr G Kummer −−−−−→ factorization gerbes on Gr G Reich −−−→ multiplicative gerbes on the loop group Tr(Frob) −−−−−→ covering groups.

In this article, we show that for any non-isotrivial family of abelian varieties over a rational base with big monodromy, those members that have adelic Galois representation with image as large as possible form a density-1 subset. Our results can be applied to a number of interesting families of abelian varieties, such as rational families dominating the moduli of Jacobians of hyperelliptic curves, trigonal curves, or plane curves. As a consequence, we prove that for any dimension g ≥ 3, there are infinitely many abelian varieties over Q with adelic Galois representation having image equal to all of GSp 2g ( Z).

Let X be a smooth algebraic variety over k. We prove that any flat quasicoherent sheaf on Ran(X) canonically acquires a D-module structure. In addition, we prove that, if the geometric fiber X k is connected and admits a smooth compactification, then any line bundle on S × Ran(X) is pulled back from S, for any locally Noetherian k-scheme S. Both theorems rely on a family of results which state that the (partial) limit of an n-excisive functor defined on the category of pointed finite sets is trivial.