TY - JOUR
T1 - Specialization of Mordell-Weil ranks of abelian schemes over surfaces to curves
AU - Keller, Timo
N1 - Publisher Copyright:
© 2023 World Scientific Publishing Company.
PY - 2023/3/27
Y1 - 2023/3/27
N2 - Using the Shioda-Tate theorem and an adaptation of Silverman's specialization theorem, we reduce the specialization of Mordell-Weil ranks for abelian varieties over fields finitely generated over infinite finitely generated fields k to the specialization theorem for Néron-Severi ranks recently proved by Ambrosi in positive characteristic. More precisely, we prove that after a blow-up of the base surface S, for all vertical curves Sx of a fibration S → U ⊆Pk1 with x from the complement of a sparse subset of |U|, the Mordell-Weil rank of an abelian scheme over S stays the same when restricted to Sx.
AB - Using the Shioda-Tate theorem and an adaptation of Silverman's specialization theorem, we reduce the specialization of Mordell-Weil ranks for abelian varieties over fields finitely generated over infinite finitely generated fields k to the specialization theorem for Néron-Severi ranks recently proved by Ambrosi in positive characteristic. More precisely, we prove that after a blow-up of the base surface S, for all vertical curves Sx of a fibration S → U ⊆Pk1 with x from the complement of a sparse subset of |U|, the Mordell-Weil rank of an abelian scheme over S stays the same when restricted to Sx.
KW - abelian schemes over higher-dimensional bases
KW - rational points
KW - Specialization of Mordell-Weil ranks
KW - specialization of Néron-Severi groups
UR - http://www.scopus.com/inward/record.url?scp=85151846299&partnerID=8YFLogxK
U2 - 10.48550/arXiv.2301.12816
DO - 10.48550/arXiv.2301.12816
M3 - Article
AN - SCOPUS:85151846299
VL - 19
SP - 1671
EP - 1680
JO - International Journal of Number Theory
JF - International Journal of Number Theory
SN - 1793-0421
IS - 7
ER -