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Strong Mg ii and Fe ii Absorbers at 2.2 z < 6.0
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null (Ed.)Abstract We continue the study of n -dependent groups, fields and related structures, largely motivated by the conjecture that every n -dependent field is dependent. We provide evidence toward this conjecture by showing that every infinite n -dependent valued field of positive characteristic is henselian, obtaining a variant of Shelah’s Henselianity Conjecture in this case and generalizing a recent result of Johnson for dependent fields. Additionally, we prove a result on intersections of type-definable connected components over generic sets of parameters in n -dependent groups, generalizing Shelah’s absoluteness of $$G^{00}$$ in dependent theories and relative absoluteness of $$G^{00}$$ in $$2$$ -dependent theories. In an effort to clarify the scope of this conjecture, we provide new examples of strictly $$2$$ -dependent fields with additional structure, showing that Granger’s examples of non-degenerate bilinear forms over dependent fields are $$2$$ -dependent. Along the way, we obtain some purely model-theoretic results of independent interest: we show that n -dependence is witnessed by formulas with all but one variable singletons; provide a type-counting criterion for $$2$$ -dependence and use it to deduce $$2$$ -dependence for compositions of dependent relations with arbitrary binary functions (the Composition Lemma); and show that an expansion of a geometric theory T by a generic predicate is dependent if and only if it is n -dependent for some n , if and only if the algebraic closure in T is disintegrated. An appendix by Martin Bays provides an explicit isomorphism in the Kaplan-Scanlon-Wagner theorem.more » « less
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We prove a Tannakian form of Drinfeld's lemma for isocrystals on a variety over a finite field, equipped with actions of partial Frobenius operators. This provides an intermediate step towards transferring V. Lafforgue's work on the Langlands correspondence over function fields from$$\ell$$-adic to$$p$$-adic coefficients. We also discuss a motivic variant and a local variant of Drinfeld's lemma.more » « less
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Abstract In this paper, we describe a certain kind of q -connections on a projective line, namely Z -twisted ( G , q ) {(G,q)} -opers with regular singularities using the language of generalized minors. In part one we explored the correspondence between these q -connections and 𝑄𝑄 \mathit{QQ} -systems/Bethe Ansatz equations. Here we associate to a Z -twisted ( G , q ) {(G,q)} -oper a class of meromorphic sections of a G -bundle, satisfying certain difference equations, which we refer to as ( G , q ) {(G,q)} -Wronskians. Among other things, we show that the 𝑄𝑄 \mathit{QQ} -systems and their extensions emerge as the relations between generalized minors, thereby putting the Bethe Ansatz equations in the framework of cluster mutations known in the theory of double Bruhat cells.more » « less
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.3847/1538-4357/abc6ff
