Counterexamples to the Hasse principle
نویسنده
چکیده
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(kv) is automatically non-empty for all but finitely many places v, which can be determined: this follows from the Weil conjectures. It is therefore a finite (and usually very straightforward) process to check whether X(kv) is non-empty for all v. For some families of varieties, this is enough to ensure that X(k) is nonempty. For example:
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