Counting Rational Points on Smooth Cyclic Covers
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چکیده
A conjecture of Serre concerns the number of rational points of bounded height on a finite cover of projective space Pn−1. In this paper, we achieve Serre’s conjecture in the special case of smooth cyclic covers of any degree when n ≥ 10, and surpass it for covers of degree r ≥ 3 when n > 10. This is achieved by a new bound for the number of perfect r-th power values of a polynomial with nonsingular leading form, obtained via a combination of an r-th power sieve and the q-analogue of van der Corput’s method.
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تاریخ انتشار 2012