منابع مشابه
Counterexamples 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 کاملPowers of Two Modulo Powers of Three
Since 2 is a primitive root of 3 for each positive integer m, the set of points {(n, 2 mod 3) : n > 0}, viewed as a subset of Z>0×Z>0 is bi-periodic, with minimal periods φ(3) (horizontally) and 3 (vertically). We show that if one considers the classes of n modulo 6, one obtains a finer structural classification. This result is presented within the context of the question of strong normality of...
متن کاملCongruences modulo Prime Powers
Let p be any prime, and let α and n be nonnegative integers. Let r ∈ Z and f (x) ∈ Z[x]. We establish the congruence p deg f k≡r (mod p α) n k (−1) k f k − r p α ≡ 0 mod p ∞ i=α ⌊n/p i ⌋ (motivated by a conjecture arising from algebraic topology), and obtain the following vast generalization of Lucas' theorem: If α > 1 and l, s, t are nonnegative integers with s, t < p, then 1 ⌊n/p α−1 ⌋! k≡r (...
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: 1999
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa-87-3-269-285