1786 - 0091 on the Mappings of Elliptic Curves Defined over Q Into
نویسنده
چکیده
Let E be an elliptic curve defined over Q given by an affine Weierstrass equation of the form (1) E : y = x + 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 Fp. Considering an infinite sequence of elliptic curves Ẽp, we map the point (x, y) of them into the unit square [0, 1) via the mapping (x, y) 7→ ( 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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