منابع مشابه
A Note on Pépin’s counter examples to the Hasse principle for curves of genus 1
In a series of articles published in the C.R. Paris more than a century ago, T. Pépin announced a list of “theorems” concerning the solvability of diophantine equations of the type ax + by = z. In this article, we show how to prove these claims using the structure of 2-class groups of imaginary quadratic number fields. We will also look at a few related results from a modern point of view.
متن کاملOn the Hasse Principle for Shimura Curves
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...
متن کاملCounterexamples to the Hasse Principle
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.
متن کاملCounterexamples to the Hasse principle
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...
متن کاملMore cubic surfaces violating the Hasse principle
We generalize L. J. Mordell’s construction of cubic surfaces for which the Hasse principle fails.
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ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 1991
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa-59-2-145-147