منابع مشابه
On 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 کاملAn “anti-hasse Principle” for Prime Twists
Given an algebraic curve C/Q having points everywhere locally and endowed with a suitable involution, we show that there exists a positive density family of prime quadratic twists of C violating the Hasse principle. The result applies in particular to wN -Atkin-Lehner twists of most modular curves X0(N) and to wp-Atkin-Lehner twists of certain Shimura curves XD+.
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 کاملA Hasse Principle for Quadratic Forms over Function Fields
We describe the classical Hasse principle for the existence of nontrivial zeros for quadratic forms over number fields, namely, local zeros over all completions at places of the number field imply nontrivial zeros over the number field itself. We then go on to explain more general questions related to the Hasse principle for nontrivial zeros of quadratic forms over function fields, with referen...
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ژورنال
عنوان ژورنال: International Journal of Number Theory
سال: 2013
ISSN: 1793-0421,1793-7310
DOI: 10.1142/s1793042113500747