On the mappings of elliptic curves defined over ℚ into [0, 1

Csajbók, Zoltán Ernő (2007) On the mappings of elliptic curves defined over ℚ into [0, 1. ACTA MATHEMATICA ACADEMIAE PAEDAGOGICAE NYÍREGYHÁZIENSIS, 23 (2). pp. 115-123. ISSN 0866-0174

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Abstract

Let E be an elliptic curve defined over ℚ given by an affine Weierstrass equation of the form (1) E : y2 = x3 + ax + b (a, b ∈ ℤ, x, y ∈ ℚ). Reducing the elliptic curve (1) modulo a sufficiently large prime p, we obtain an elliptic curve Ẽp over double-struck F signp. Considering an infinite sequence of elliptic curves Ẽp, we map the point (x,y) of them into the unit square [0, 1)2 via the mapping (x, y) → (x/p, y/p). We prove that the obtained cumulative point set contains a point sequence aligning a line when E/ℚ has an integral point, and point sequences aligning lines of well defined number when E/ℚ has a rational point. In both cases, these lines contain infinitely many points being strictly monotone increasing or decreasing according to the L∞ norm, and these monotone point sequences converge to well defined points.

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