Higher reciprocity laws and rational points (Q2790621)

This paper establishes an obstruction theory for rational points on varieties defined over function fields of two-dimensional schemes of some geometric type. The theory is applied to obtain interesting counterexamples to the local-global principle for torsors and Galois cohomology of finite modules.NEWLINENEWLINEThe base field the authors mainly work with is either the function field \(K\) of an algebraic curve over the fraction field of a Henselian, excellent discrete valuation ring \(R\) (the \textit{semi-global} case), or the fraction field \(K\) of a two-dimensional, Henselian, excellent, local domain \(R\) (the \textit{local} case). In both cases, the field \(K\) can be realized as the function field of a regular model \(\mathcal{X}\) over \(R\), i.e., a two-dimensional regular scheme \(\mathcal{X}\) equipped with a surjective projective morphism to \(\mathrm{Spec}(R)\). Over such a field \(K\), the two guiding questions, which arise as natural analogues of well known results in the global field case, are the following:NEWLINENEWLINEQuestion 1. Let \(G\) be a connected linear algebraic group over \(K\). For every discrete valuation \(v\) of \(K\), let \(K_v\) denote the Henselisation (or completion) of \(K\) at \(v\). Suppose that a \(G\)-torsor has a \(K_v\)-point for every \(v\). Does it necessarily have a \(K\)-point?NEWLINENEWLINEQuestion 2. Let \(\mu\) be a finite Galois module over \(K\) and let \(i\geq 1\) be an integer. Does the natural map NEWLINE\[NEWLINE H^i(K,\,\mu)\longrightarrow \prod_vH^i(K_v,\,\mu) NEWLINE\]NEWLINE have trivial kernel?NEWLINENEWLINEThe introduction of the paper provides a very nice review of the background and related results about these questions. For Question 1, a positive answer was previously obtained in quite a number of cases. Consider for example the case where the residue field \(k\) of the local ring \(R\) is algebraically closed of characteristic 0. Then the field \(K\) is perfect of cohomological dimension 2. So by Serre's conjecture II (which has been proved for the field \(K\), see [the first author et al., in: Proceedings of the international colloquium on algebra, arithmetic and geometry, Mumbai, India, January 4--12, 2000. Part I and II. New Delhi: Narosa Publishing House. 185--217 (2002; Zbl 1055.14019)], \(H^1(K,\,G)\) is trivial when \(G\) is semisimple and simply connected. The answer to Question 1 is thus positive in that case. Another case is with \(G\) (connected) and rational over \(K\) (see the first author et al. [Duke Math. J. 121, No. 2, 285--341 (2004; Zbl 1129.11014); \textit{D. Harbater} et al., Am. J. Math. 137, No. 6, 1559--1612 (2015; Zbl 1348.11036)]).NEWLINENEWLINEAs for Question 2, the answer was already known to be negative for \(i=1\) and \(\mu=\mathbb{Z}/2\). On the other hand, the answer is yes for \(i\geq 2\) and \(\mu=\mu_n^{\otimes (i-1)}\) with \(n\) invertible in the residue field \(k\) (see [\textit{D. Harbater} et al., Comment. Math. Helv. 89, No. 1, 215--253 (2014; Zbl 1332.11046); \textit{Y. Hu}, ``A cohomological Hasse principle over two-dimensional local rings'', Preprint, \url{arXiv:1401.7782}]). This in turn answers Question 1 affirmatively for some (e.g., quasi-split, of classical type) semisimple simply connected groups, even for an arbitrary residue field \(k\).NEWLINENEWLINENevertheless, all the earlier results leave the following questions open: Does the local-global principle in Question 1 fail for some reductive, or even semisimple (but not simply connected) groups? For \(i=2\), is there a finite module for which Question 2 has a negative answer?NEWLINENEWLINEThe paper under review solves these questions by constructing counterexamples, which turns out to exist already in the case where \(k\) is algebraically closed of characteristic 0. A key machinery is a new type of obstruction to the existence of rational points over regular excellent schemes of arbitrary dimension. This obstruction theory, called \textit{reciprocity obstruction}, was proposed by the first author several years ago and is derived from Kato's version of the Bloch-Ogus complex.NEWLINENEWLINEUsing suitable regular models \(\mathcal{X}/R\) for the field \(K\), the authors first give an explicit counterexample to the local-global principle involved in Question 1. They choose \(G\) to be a normic torus defined by an equation of the form NEWLINE\[NEWLINE (X_1^2-aY_1^2)(X_2^2-bY_2^2)(X_3^2-abY_3^2)=1\;. NEWLINE\]NEWLINEAs was shown by the first author, suitable values of \(a,\,b\in K^*\) can be chosen so that torsors of the torus have a nonzero unramified Brauer group. Thus, the degree \(2\) case of the reciprocity obstruction can be used to prove the non-existence of rational points, just as what can be done with the Brauer-Manin obstruction over global fields. Compared with the work of Harbater et al. [loc. cit., Zbl 1348.11036], it is worth noticing that in this example the torus \(G\) is not \(K\)-rational and the construction does rely on the bad combinatorial behavior of the closed fiber of the model \(\mathcal{X}\). In Proposition 5.11, it is explained that this example also violates the local-global principles arising from the patching setup of Harbater, Hartmann and Krashen.NEWLINENEWLINEWith the aforementioned counterexample and using some classic techniques, the authors further deduce a negative answer to Question 2 for \(i=2\), which in return yields a counterexample to Question 1 with \(G\) semisimple (not simply connected).NEWLINENEWLINEFinally, it is also found that (at least in the local case) when the residue field \(k\) is separably closed of characteristic unequal to 2, the reciprocity obstruction used in this paper is the only obstruction (to the existence of rational points) for torsors under the above mentioned normic torus. The authors are led to ask whether this phenomenon is still true in some more general context. This last question has been answered affirmatively in a recent work of \textit{D. Izquierdo} [``Dualité et principe local-global sur des corps locaux de dimension 2'', Preprint, \url{arXiv:1605.0128}]