Journal of the Ramanujan Mathematical Society

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Rational Points on Diagonal Cubic Surfaces


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1 Mathematical Institute, Tohoku University, Sendai, Miyagi-980-8578, Japan
     

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We show under the assumption that the Tate-Shafarevich group of any elliptic curve over ℚ is finite that the cubic surface x31+p1p2x32+p2p3x33+p3p1x34=0 over ℚ has a rational point, where p1, p2 and p3 are rational primes, each congruent to either 2 or 5 modulo 9.
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  • Rational Points on Diagonal Cubic Surfaces

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Authors

Kazuki Sato
Mathematical Institute, Tohoku University, Sendai, Miyagi-980-8578, Japan

Abstract


We show under the assumption that the Tate-Shafarevich group of any elliptic curve over ℚ is finite that the cubic surface x31+p1p2x32+p2p3x33+p3p1x34=0 over ℚ has a rational point, where p1, p2 and p3 are rational primes, each congruent to either 2 or 5 modulo 9.