Narrow Results

Let K_i be a number field for all i ∈ ℤ_>0 and let ℰ be a family of elliptic curves containing infinitely many members defined over K_i for all i. Fix a rational prime p. We give sufficient conditions for the existence of an integer i₀ such that, for all i > i₀ and all elliptic curve E ∈ ℰ having good reduction at all 𝔭 | p in K_i, we have that E has good ordinary reduction at all primes 𝔭 | p. We illustrate our criteria by applying it to certain Frey curves in [Recipes to Fermat-type equations of the form x^r + y^r = Cz^p, to appear in Math. Z.; http://arXiv.org/abs/1203.3371] attached to Fermat-type equations of signature (r, r, p).

We prove that the equation ${(x-y)}^4+x^4+{(x+y)}^4=z^n$ has no integer solutions $x,y,z$ with $gcd(x,y)=1$ for all integers $n>1$. We mainly use a modular approach with two Frey $\mathbb{Q}$-curves defined over the field $\mathbb{Q}(\sqrt{30})$.

In the modern theory of elliptic curves, one of the important problems is the determination of the number of rational points on an elliptic curve. The Mordel–Weil theorem [T. Shioda, On the Mordell–Weil lattices, Comment. Math. University St. Paul.39(2) (1990) 211–240] points out that the elliptic curve defined above the rational points is generated by a finite group. Despite the knowledge that an elliptic curve has a final number of rational points, it is still difficult to determine their number and the way how to determine them. The greatest progress was achieved by Birch and Swinnerton–Dyer conjecture, which was included in the Millennium Prize Problems [A. Wiles, The Birch and Swinnerton–Dyer conjecture, The Millennium Prize Problems (American Mathematical Society, 2006), pp. 31–44]. This conjecture uses methods of the analytical theory of numbers, while the current knowledge corresponds to the assumptions of the conjecture but has not been proven to date. In this paper, we focus on using a tangent line and the osculating circle for characterizing the rational points of the elliptical curve, which is the greatest benefit of the contribution. We use a different view of elliptic curves by using Minkowki’s theory of number geometry [H. F. Blichfeldt, A new principle in the geometry of numbers, with some applications, Trans. Amer. Math. Soc.15(3) (1914) 227–235; V. S. Miller, Use of elliptic curves in cryptography, in Proc. Advances in Cryptology — CRYPTO ’85, Lecture Notes in Computer Science, Vol. 218 (Springer, Berlin, Heidelberg, 1985), pp. 417–426; E. Bombieri and W. Gubler, Heights in Diophantine Geometry, Vol. 670, 1st edn. (Cambridge University Press, 2007)].