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A Note on Artin's Diophantine Conjecture
Published online by Cambridge University Press: 20 November 2018
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A well known theorem of Hasse [1] says that every quadratic form in at least 5 variables over the field Qp of p-adic numbers has a nontrivial zero. This fact has led Artin to make the conjecture
(C): "Every form over Qp of degree d in n > d2 variables has a non-trivial zero." However, a counterexample has been provided by Terjanian [2] in the case d=4.
The case d=2 is distinguished by the fact that every quadratic form may be "diagonalized", i.e., assumed to be of the type Σ aiX2i. One is therefore led to the weaker conjecture
(C): "Every form f= Σ aiXdi over Qp in n > d2 variables has a nontrivial zero in Qp,"
which still generalizes Hasse's theorem.
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- Research Article
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- Copyright
- Copyright © Canadian Mathematical Society 1970
References
1.
Hasse, H., Darstellbarkeit von Zahlen durch Quadratische Formen, J. f. reine u. angew. Math. 153 (1923), 113-130.Google Scholar
2.
Terjanian, G., Un contre-exemple à une conjecture d' Artin, Comptes Rendus de l' Acad. Sci. Paris
262 (1966), A612.Google Scholar
3.
Chevalley, C., Démonstration d'une hypothèse de M. Artin, Abh. Math. Sem. Hambur.
11 (1935), 73-75.Google Scholar
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