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A rational triangle is a triangle with sides of rational lengths. In this short note, we prove that there exists a unique pair of a rational right triangle and a rational isosceles triangle which have the same perimeter and the same area. In the proof, we determine the set of rational points on a certain hyperelliptic curve by a standard but sophisticated argument which is based on the 2-descent on its Jacobian variety and Coleman's theory of p-adic abelian integrals.

In this article, we introduce a systematic and uniform construction of nonsingular plane curves of odd degrees n ≥ 5 which violate the local-global principle. Our construction works unconditionally for n divisible by p 2 for some odd prime number p. Moreover, our construction also works for n divisible by some p ≥ 5 which satisfies a conjecture on a p-adic property of the fundamental unit of Q(p 1/3 ). This conjecture is a natural cubic analogue of the classical Ankeny-Artin-Chowla-Mordell conjecture for Q(p 1/2 ) and easily verified numerically.

In this paper, we construct some families of infinitely many hyperelliptic curves of genus 2 with exactly two rational points. In the proof, we first show that the Mordell-Weil ranks of these hyperelliptic curves are 0 and then determine the sets of rational points by using the Lutz-Nagell type theorem for hyperelliptic curves which was proven by Grant.Contents2), p ≡ −3 (mod 8) 18 3. Application of the Lutz-Nagell type theorem for hyperelliptic curves 21 4. Concluding remarks 23 Appendix A. 23 References 24

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