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Register a new userLocal-global principles for constant reductive groups over semi-global fields
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We study local-global principles for torsors under reductive linear algebraic groups over semi-global fields; i.e., over one variable function fields over complete discretely valued fields. We provide conditions on the group and the semiglobal field under which the local-global principle holds, and we compute the obstruction to the local-global principle in certain classes of examples. Using our description of the obstruction, we give the first example of a semisimple simply connected group over a semi-global field where the local-global principle fails. Our methods include patching and R-equivalence.
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In this paper we study local-global principles for tori over semi-global fields, which are one variable function fields over complete discretely valued fields. In particular, we show that for principal homogeneous spaces for tori over the underlying discrete valuation ring, the obstruction to a local-global principle with respect to discrete valuations can be computed using methods coming from patching. We give a sufficient condition for the vanishing of the obstruction, as well as examples were the obstruction is nontrivial or even infinite. A major tool is the notion of a flasque resolution of a torus.
This is the final version, to appear in Commentarii Mathematici Helvetici.
We prove the failure of the local-global principle, with respect to all discrete valuations, for isotropy of quadratic forms over a rational function field of transcendence degree at least 2 over the complex numbers. Our construction involves the generalized Kummer varieties considered by Borcea and Cynk--Hulek.
Let K be a global field and f in K[X] be a polynomial. We present an efficient algorithm which factors f in polynomial time.
We construct the Frobenius structure on a rigid connection $mathrm{Be}_{check{G}}$ on $mathbb{G}_m$ for a split reductive group $check{G}$ introduced by Frenkel-Gross. These data form a $check{G}$-valued overconvergent $F$-isocrystal $mathrm{Be}_{check{G}}^{dagger}$ on $mathbb{G}_{m,mathbb{F}_p}$, which is the $p$-adic companion of the Kloosterman $check{G}$-local system $mathrm{Kl}_{check{G}}$ constructed by Heinloth-Ng^o-Yun. By exploring the structure of the underlying differential equation, we calculate the monodromy group of $mathrm{Be}_{check{G}}^{dagger}$ when $check{G}$ is almost simple (which recovers the calculation of monodromy group of $mathrm{Kl}_{check{G}}$ due to Katz and Heinloth-Ng^o-Yun), and establish functoriality between different Kloosterman $check{G}$-local systems as conjectured by Heinloth-Ng^o-Yun. We show that the Frobenius Newton polygons of $mathrm{Kl}_{check{G}}$ are generically ordinary for every $check{G}$ and are everywhere ordinary on $|mathbb{G}_{m,mathbb{F}_p}|$ when $check{G}$ is classical or $G_2$.
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