We report on computer search for generalized Gosper curve for 37 < N < 61, where N is the degree of the generalized Gosper curve. From the results of the computer search and some geometrical insight, we conjecture that the degree N satisfies N = 6n+1. We investigate the existence of infinite series of generalized Gosper curves. We show how to generate these series and introduce two new methods,...

Let E be an elliptic curve over FqðTÞ with conductor N N: Let Y : X0ðNÞ-E be the modular parametrization by the Drinfeld modular curve of level N: Assuming that E is a strong Weil curve we prove upper and lower bounds on deg Y: These bounds are the analogs of well-known (partially conjectural) bounds in the case of rational numbers. r 2002 Elsevier Science (USA). All rights reserved.

Let $E$ be an elliptic curve over $Bbb{Q}$ with the given Weierstrass equation $ y^2=x^3+ax+b$. If $D$ is a squarefree integer, then let $E^{(D)}$ denote the $D$-quadratic twist of $E$ that is given by $E^{(D)}: y^2=x^3+aD^2x+bD^3$. Let $E^{(D)}(Bbb{Q})$ be the group of $Bbb{Q}$-rational points of $E^{(D)}$. It is conjectured by J. Silverman that there are infinitely many primes $p$ for which $...