# zbMATH — the first resource for mathematics

The Diophantine equation $$X^3=u+27v$$ over real quadratic fields. (English) Zbl 1227.11052
Let $$k={\mathbb Q}(\sqrt{6})$$ or $$k={\mathbb Q}(\sqrt{3p})$$, where $$p$$ is a prime number, $$p\neq 3$$, $$p\equiv 3(\bmod 4)$$, $${\mathcal O}_k$$ the integers of $$k$$, $${\mathcal O}_k^\times$$ the units. The author proves that if the equation $$X^3=u+27v$$ has solutions $$X\in {\mathcal O}_k$$, $$u,v\in{\mathcal O}_k^\times$$, then $$k={\mathbb Q}(\sqrt{6})$$ or $$k={\mathbb Q}(\sqrt{33})$$, and the only solutions are $(X,u,v)=(w_1(4\pm \sqrt{6}),w_1^3,w_1^3(5\pm 2\sqrt{6}))$ for any $$w_1\in{\mathcal O}_{{\mathbb Q}(\sqrt{6})}^\times$$, or $(X,u,v)=(w_2(5\pm \sqrt{33}),-w_2^3,w_2^3(23\pm 4\sqrt{33}))$ for any $$w_2\in{\mathcal O}_{{\mathbb Q}(\sqrt{33})}^\times$$.
##### MSC:
 11D25 Cubic and quartic Diophantine equations 11R11 Quadratic extensions
##### Keywords:
real quadratic field; cubic Diophantine equation
Full Text:
##### References:
  A. Fröhlich, Central extensions, Galois groups, and ideal class groups of number fields , Contemp. Math. 24, American Mathematical Society, 1983. · Zbl 0519.12001  T. Kagawa, Determination of elliptic curves with everywhere good reduction over $$\Q(\sqrt{37})$$, Acta Arith., 83 (1998), 253-269. · Zbl 0915.11033 · eudml:207122  T. Kagawa, Nonexistence of elliptic curves having everywhere good reduction and cubic discriminant, Proc. Japan Acad., 76 , Ser.,A (2000), 141-142. · Zbl 0991.11029 · doi:10.3792/pjaa.76.141  T. Kagawa, Determination of elliptic curves with everywhere good reduction over real quadratic fields $$\Q(\sqrt{3p})$$, Acta Arith., 96 (2001), 231-245. · Zbl 0977.11024 · doi:10.4064/aa96-3-4 · eudml:207122  T. Kagawa, The Diophantine equation $$X^3=u+v$$ over real quadratic fields, in, preparation. · Zbl 1228.11039  M. Kida, Arithmetic of abelian varieties under field extensions , dissertation, Johns Hopkins, 1994.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.