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Visible Points on Curves over Finite Fields

Igor E. Shparlinski, José Felipe Voloch (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

For a prime p and an absolutely irreducible modulo p polynomial f(U,V) ∈ ℤ[U,V] we obtain an asymptotic formula for the number of solutions to the congruence f(x,y) ≡ a (mod p) in positive integers x ≤ X, y ≤ Y, with the additional condition gcd(x,y) = 1. Such solutions have a natural interpretation as solutions which are visible from the origin. These formulas are derived on average over a for a fixed prime p, and also on average over p for a fixed integer a.

Visible Points on Modular Exponential Curves

Tsz Ho Chan, Igor E. Shparlinski (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

We obtain an asymptotic formula for the number of visible points (x,y), that is, with gcd(x,y) = 1, which lie in the box [1,U] × [1,V] and also belong to the exponential modular curves y ≡ a g x ( m o d p ) . Among other tools, some recent results of additive combinatorics due to J. Bourgain and M. Z. Garaev play a crucial role in our argument.

Volcanoes of l-isogenies of elliptic curves over finite fields: The case l=3.

Josep M. Miret Biosca, Daniel Sadornil Renedo, Juan Tena Ayuso, Rosana Tomàs, Magda Valls Marsal (2007)

Publicacions Matemàtiques

This paper is devoted to the study of the volcanoes of ℓ-isogenies of elliptic curves over a finite field, focusing on their height as well as on the location of curves across its different levels. The core of the paper lies on the relationship between the ℓ-Sylow subgroup of an elliptic curve and the level of the volcano where it is placed. The particular case ℓ = 3 is studied in detail, giving an algorithm to determine the volcano of 3-isogenies of a given elliptic curve. Experimental results...

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