TAILIEUCHUNG - Đề tài " The number of extensions of a number field with fixed degree and bounded discriminant "

We give an upper bound on the number of extensions of a fixed number field of prescribed degree and discriminant ≤ X; these bounds improve on work of Schmidt. We also prove various related results, such as lower bounds for the number of extensions and upper bounds for Galois extensions. | Annals of Mathematics The number of extensions of a number field with fixed degree and bounded discriminant By Jordan S. Ellenberg and Akshay Venkatesh Annals of Mathematics 163 2006 723 741 The number of extensions of a number field with fixed degree and bounded discriminant By Jordan S. Ellenberg and ÀKSHAy Venkatesh Abstract We give an upper bound on the number of extensions of a fixed number field of prescribed degree and discriminant X these bounds improve on work of Schmidt. We also prove various related results such as lower bounds for the number of extensions and upper bounds for Galois extensions. 1. Introduction Let K be a number field and let NK n X be the number of number fields L always considered up to K-isomorphism such that L K n and Nq Dl k X. Here DL K is the relative discriminant of L K and Nq is the norm on ideals of K valued in positive integers. DL DL q will refer to discriminant over Q. A folk conjecture possibly due to Linnik asserts that NK n X CK nX n fixed X to . This conjecture is trivial when n 2 it has been proved for n 3 by Davenport and Heilbronn 7 in case K Q and by Datskovsky and Wright in general 6 and for n 4 5 and K Q by Bhargava 3 2 . A weaker version of the conjecture for n 5 was also recently established by Kable and Yukie 11 . These beautiful results are proved by methods which seem not to extend to higher n. The best upper bound for general n is due to Schmidt 18 who showed NK n X X n 2 4 where the implied constant depends on K and n. We refer to 4 for a survey of results. In many cases it is easy to show that NK n X is bounded below by a constant multiple of X for instance if n is even simply consider the set of The first author was partially supported by NSA Young Investigator Grant MDA905-02-1-0097. The second author was partially supported by NSF Grant DMS-0245606. 724 JORDAN S. ELLENBERG AND AKSHAY VENKATESH quadratic extensions of a fixed L0 K of degree n 2. For the study of lower bounds it is therefore more .

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