A pr 2 00 8 K 3 surfaces with Picard rank 20
نویسنده
چکیده
We determine all complex K3 surfaces with Picard rank 20 over Q. Here the NéronSeveri group has rank 20 and is generated by divisors which are defined over Q. Our proof uses modularity, the Artin-Tate conjecture and class group theory. With different techniques, the result has been established by Elkies to show that Mordell-Weil rank 18 over Q is impossible for an elliptic K3 surface. We also apply our methods to general singular K3 surfaces, i.e. with Néron-Severi group of rank 20, but not necessarily generated by divisors over Q.
منابع مشابه
K 3 surfaces with Picard rank 20 over Q
We compute all K3 surfaces with Picard rank 20 over Q. Our proof uses modularity, the Artin-Tate conjecture and class group theory. With different techniques, the result has been established by Elkies to show that Mordell-Weil rank 18 over Q is impossible for an elliptic K3 surface. We also apply our methods to general singular K3 surfaces, i.e. with geometric Picard rank 20, but not necessaril...
متن کاملar X iv : 0 80 4 . 15 58 v 2 [ m at h . N T ] 1 7 Ju n 20 08 K 3 surfaces with Picard rank 20
We determine all complex K3 surfaces with Picard rank 20 over Q. Here the NéronSeveri group has rank 20 and is generated by divisors which are defined over Q. Our proof uses modularity, the Artin-Tate conjecture and class group theory. With different techniques, the result has been established by Elkies to show that Mordell-Weil rank 18 over Q is impossible for an elliptic K3 surface. We also a...
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Frequentely it happens that isogenous (in the sense of Mukai) K3 surfaces are partners of each other and sometimes they are even isomorphic. This is due, in some cases, to the (too high, e.g. bigger then or equal to 12)) rank of the Picard lattice as showed by Mukai in [13]. In other cases this is due to the structure of the Picard lattice and not only on its rank. This is the case, for example...
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In general, not much is known about the arithmetic of K3 surfaces. Once the geometric Picard number, which is the rank of the Néron-Severi group over an algebraic closure of the base field, is high enough, more structure is known and more can be said. However, until recently not a single K3 surface was known to have geometric Picard number one. We give explicit examples of such surfaces over th...
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