Twists of elliptic curves with a large set of integral points over Fq(t).

摘要

For q ≡ 3 mod 4, we show that the quadratic twist ( tqn -- t)y2 = x3 -- x has at least tau odd(n) infinity--integral points over Fq&parl0;t&parr0; in different Frobenius orbit, where tauodd( n) is the number of odd divisors of an integer n. We also show that the same holds true if we consider cubic twists of y2 = x3 + 1 in the supersingular case, i.e., q ≡ 2 mod 3.;These examples allow us to show that the conjecture of Lang-Vojta concerning the behavior of integral points in varieties of log-general type cannot be readily transported to the function field case. Also for q ≡ 2 mod 3 they show that there are cubic twists of elliptic curves over Fq&parl0;t&parr0; with arbitrarily large rank.