The Hasse norm principle for abelian extensions
نویسندگان
چکیده
منابع مشابه
The Proportion of Failures of the Hasse Norm Principle
For any number field we calculate the exact proportion of rational numbers which are everywhere locally a norm but not globally a norm from the number field.
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Let C be an algebraic curve defined over a number field K, of positive genus and without K-rational points. We conjecture that there exists some extension field L over which C violates the Hasse principle, i.e., has points everywhere locally but not globally. We show that our conjecture holds for all but finitely many Shimura curves of the form XD 0 (N)/Q or X D 1 (N)/Q, where D > 1 and N are c...
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This article explains the Hasse principle and gives a self-contained development of certain counterexamples to this principle. The counterexamples considered are similar to the earliest counterexample discovered by Lind and Reichardt. This type of counterexample is important in the theory of elliptic curves: today they are interpreted as nontrivial elements in Tate– Shafarevich groups.
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In both of these examples, we have proved that X(Q) = ∅ by showing that X(Qv) = ∅ for some place v. In the first case it was v = ∞, the real place. In the second case we showed that X(Q2) was empty: the argument applies equally well to a supposed solution over Q2. Given a variety X over a number field k and a place v of k, it is a finite procedure to decide whether X(kv) is empty. Moreover, X(k...
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We generalize L. J. Mordell’s construction of cubic surfaces for which the Hasse principle fails.
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ژورنال
عنوان ژورنال: American Journal of Mathematics
سال: 2018
ISSN: 1080-6377
DOI: 10.1353/ajm.2018.0048