Counting rational points on smooth cyclic covers
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> 1...
Main Authors: | , |
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Format: | Journal article |
Language: | English |
Published: |
2012
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Summary: | 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. © 2012 Elsevier Inc. |
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