Search ResultsShowing 1-20 of 23
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arXiv:2410.15598 (Published 2024-10-21)
Rost injectivity for classical groups over function fields of curves over local fields
Comments: 23 pagesLet F be a complete discretely valued field with residue field a global field or a local field with no real orderings. Let G be an absolutely simple simply connected group of outer type A_n. If 2 and the index of the underlying algebra of G are coprime to the characteristic of the residue field of F, then we prove that the Rost invariant map from the first Galois cohomology set of G to the degree three Galois cohomology group is injective. Let L be the function field of a curve over a local field K and G an absolutely simple simply connected linear algebraic group over L of classical type. Suppose that the characteristic of the residue field of K is a good prime for G. As a consequence of our result and some known results we conclude that the Rost invariant of G is injective.
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arXiv:2406.17355 (Published 2024-06-25)
Local-global principle for over semiglobal fields
We compare different local-global principles for torsors under a reductive group G defined over a semiglobal field F. In particular if the F-group G s a retract rational F-variety, we prove that the local global principle holds for the completions with respect to divisorial valuations of F.
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arXiv:2312.03934 (Published 2023-12-06)
Period-index in top cohomology over semiglobal fields
Comments: Comments are welcome!We prove a common slot lemma for symbols in top cohomology classes over semiglobal fields. Furthermore, we prove that period and index agree for general top cohomology classes over such fields. We discuss applications to quadratic forms and related open problems.
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arXiv:2301.07572 (Published 2023-01-18)
A local-global principle for twisted flag varieties
We prove a local-global principle for twisted flag varieties over a semiglobal field.
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arXiv:2201.12780 (Published 2022-01-30)
Period -Index problem for hyperelliptic curves
Let $C$ be a smooth projective curve of genus 2 over a number field $k$ with a rational point. We prove that the index and exponent coincide for elements in the 2-torsion of $\Sha(Br(C))$. In the appendix, an isomorphism of the moduli space of rank 2 stable vector bundles with odd determinant on a smooth projective hyperelliptic curve $C$ of genus $g$ with a rational point over any field of characteristic not two with the Grassmannian of $(g-1)$-dimensional linear subspaces in the base locus of a certain pencil of quadrics is established, making a result of (\cite{De-Ra}) rational. We establish a twisted version of this isomorphism and we derive as a consequence a weak Hasse principle for the smooth intersection $X$ of two quadrics in ${\mathbb P}^5$ over a number field: if $X$ contains a line locally, then $X$ has a $k$-rational point.
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arXiv:2004.10357 (Published 2020-04-22)
Local-Global Principle for Unitary Groups Over Function Fields of p-adic Curves
Let K be a p-adic field and F the function field of a curve over K. Let G be a connected linear algebraic group over F of classical type. Suppose the prime p is a good prime for G. Then we prove that projective homogeneous spaces under G over F satisfy a local global principle for rational points with respect to discrete valuations of F . If G is a semisimple simply connected group over F , then we also prove that principal homogeneous spaces under G over F satisfy a local global principle for rational points with respect to discrete valuations of F.
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arXiv:2004.08850 (Published 2020-04-19)
On unramified Brauer groups of torsors over tori
In this paper we introduce a method to obtain algebraic information using arithmetic one in the study of tori and their principal homogeneous spaces. In particular, using some results of the authors with Tingyu Lee, we determine the unramified Brauer groups of some norm one tori, and their torsors.
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arXiv:1911.07666 (Published 2019-11-18)
On the Grothendieck-Serre Conjecture for Classical Groups
Comments: 29+10 pages (paper and appendix); comments are welcomeLet $(A,\sigma)$ be an Azumaya algebra with involution over a regular semilocal ring $R$ and let $\varepsilon\in \{\pm 1\}$. We prove that the Gersten-Witt complex associated to $(A,\sigma)$ and $\varepsilon$ is exact if (i) $\dim R\leq 2$, or (ii) $\dim R\leq 3$, $\mathrm{ind}\, A\leq 2$ and $\sigma$ is of the first kind, or (iii) $\dim R\leq 4$ and $\mathrm{ind}\, A$ is odd. As a corollary, we prove the Grothendieck-Serre conjecture on principal homogeneous spaces for all forms of $\mathbf{GL}_n$, $\mathbf{Sp}_{2n}$ and $\mathbf{SO}_n$ when $\dim R\leq 2$, and all outer forms of $\mathbf{GL}_{2n+1}$ when $\dim R\leq 4$. In the process, we present a new construction of the Gersten-Witt complex, defined using explicit second-residue maps.
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arXiv:1906.10672 (Published 2019-06-25)
Local-global principles for tori over arithmetic curves
Comments: 27 pagesIn 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.
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arXiv:1901.04722 (Published 2019-01-15)
Local triviality for G-torsors
Let C $\rightarrow$ Spec(R) be a relative proper flat curve over an henselian base. Let G be a reductive C-group scheme. Under mild technical assumptions, we show that a G-torsor over C which is trivial on the closed fiber of C is locally trivial for the Zariski topology.
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arXiv:1710.03635 (Published 2017-10-10)
Local-global Galois theory of arithmetic function fields
Comments: 24 pagesWe study the relationship between the local and global Galois theory of function fields over a complete discretely valued field. We give necessary and sufficient conditions for local separable extensions to descend to global extensions, and for the local absolute Galois group to inject into the global absolute Galois group. As an application we obtain a local-global principle for the index of a variety over such a function field. In this context we also study algebraic versions of van Kampen's theorem, describing the global absolute Galois group as a pushout of local absolute Galois groups.
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arXiv:1608.04483 (Published 2016-08-16)
Cohomological invariants for G-Galois algebras and self-dual normal bases
Categories: math.NTWe define degree two cohomological invariants for G-Galois algebras over fields of characteristic not 2, and use them to give necessary conditions for the existence of a self--dual normal basis. In some cases, we show that these conditions are also sufficient.
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arXiv:1605.04728 (Published 2016-05-16)
Local-global principle for reduced norms over function fields of p-adic curves
Let F be a function field in one variable over a p-adic field and D a central division algebra over F of degree n coprime to p. We prove that Suslin invariant detects whether an element in F is a reduced norm. This leads to a local-global principle for reduced norms with respect to all discrete valuations of F.
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arXiv:1507.06277 (Published 2015-07-22)
Hasse principles for multinorm equations
Let $k$ be a global field and let $L_0$,...,$L_m$ be finite separable field extensions of $k$. In this paper, we are interested in the Hasse principle for the multinorm equation $\underset{i=0}{\overset{m}{\prod}}N_{L_i/k}(t_i)=c$. Under the assumption that $L_0$ is a cyclic extension, we give an explicit description of the Brauer-Manin obstruction to the Hasse principle. We also give a complete criterion for the Hasse principle for multinorm equations to hold when $L_0$ is a meta-cyclic extension.
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arXiv:1410.5067 (Published 2014-10-19)
Embeddings of maximal tori in classical groups and explicit Brauer-Manin obstruction
Categories: math.NTEmbeddings of maximal tori into classical groups over global fields of characteristic not 2 are the subject matter of several recent papers, with special attention to the Hasse principle. The present paper gives necessary and sufficient conditions for this embedding problem, and in particular for the Hasse principle to hold. Using work of Borovoi, this is interpreted as a Brauer-Manin type obstruction.
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arXiv:1305.3161 (Published 2013-05-14)
Hasse Principle for G-quadratic forms
Comments: To appear in Documenta MathematicaCategories: math.NTLet k be a global field of characteristic not 2. The classical Hasse-Minkowski theorem states that if two quadratic forms become isomorphic over all the completions of k, then they are isomorphic over k as well. It is natural to ask whether this is also true for G-quadratic forms, where G is a finite group. In the case of number fields the Hasse principle for G-quadratic forms does not hold in general, as shown by Jorge Morales. The aim of this paper is to study this question when k is a global field of positive characteristic. We give a sufficient criterion for the Hasse principle to hold, and also counter examples : note that these are of different nature than those for number fields.
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Lois de réciprocité supérieures et points rationnels
Comments: Final version, in French, to appear in the Transactions of the American Mathematical SocietyLet C be the complex field and K=C((x,y)) or K=C((x))(y). Let G be a connected linear algebraic group over K. Under the assumption that the K-variety G is K-rational, i.e. that the function field is purely transcendant, it was proved that a principal homogeneous space of G has a rational point over K as soon as it has one over each completion of K with respect to a discrete valuation. In this paper we show that one cannot in general do without the K-rationality assumption. To produce our examples, we introduce a new type of obstruction. It is based on higher reciprocity laws on a 2-dimensional scheme. We also produce a family of principal homogeneous spaces for which the refined obstruction controls exactly the existence of rational points.
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Hasse principle for G-trace forms
Comments: minor correctionsCategories: math.NTKeywords: g-trace forms, hasse principle, self-dual normal bases, global field, local-global principleTags: journal articleLet k be a global field of characteristic not 2. We prove a local-global principle for the existence of self-dual normal bases, and more generally for the isomorphism of G-trace forms, of G-Galois algebras over k.
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Colliot-Thélène's conjecture and finiteness of u-invariants
Comments: 25 pages, comments welcome at any timeWe show that Colliot-Th\'el\`ene's conjecture on 0-cycles of degree 1 implies finiteness for the u-invariant of the function field of a curve over a totally imaginary number field and period-index bounds for the Brauer groups of arbitrary fields of transcendence degree 1 over the rational numbers.
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arXiv:1207.4105 (Published 2012-07-17)
Quadric surface bundles over surfaces
Comments: 26 pages, comments welcomeLet T -> S be a finite flat morphism of degree two between regular integral schemes of dimension at most two (and with 2 invertible), having regular branch divisor D. We establish a bijection between Azumaya quaternion algebras on T and quadric surface bundles over S with simple degeneration along D. This is a manifestation of the exceptional isomorphism A_1^2 = D_2 degenerating to the exceptional isomorphism A_1 = B_1. In one direction, the even Clifford algebra yields the map. In the other direction, we show that the classical algebra norm functor can be uniquely extended over the discriminant divisor. Along the way, we study the orthogonal group schemes, which are smooth yet nonreductive, of quadratic forms with simple degeneration. Finally, we provide two surprising applications: constructing counter-examples to the local-global principle for isotropy, with respect to discrete valuations, of quadratic forms over surfaces; and a new proof of the global Torelli theorem for very general cubic fourfolds containing a plane.