# Journal of the Ramanujan Mathematical Society

Volume 30, Issue 3, September 2015 pp. 295–308.

Rational points on diagonal cubic surfaces

**Authors**:
Kazuki Sato

**Author institution:**Mathematical Institute, Tohoku University, Sendai, Miyagi, 980-8578, Japan

**Summary: **
We show under the assumption that the Tate-Shafarevich group of
any elliptic curve over Q is finite that the cubic surface
x1 3 + p1 p2 x2 3 + p2 p3 x3 3 + p3 p1 x4 3 = 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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