Details
| Original language | English |
|---|---|
| Pages (from-to) | 1671-1680 |
| Number of pages | 10 |
| Journal | International Journal of Number Theory |
| Volume | 19 |
| Issue number | 7 |
| Publication status | Published - 27 Mar 2023 |
Abstract
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.
Keywords
- abelian schemes over higher-dimensional bases, rational points, Specialization of Mordell-Weil ranks, specialization of Néron-Severi groups
ASJC Scopus subject areas
- Mathematics(all)
- Algebra and Number Theory
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In: International Journal of Number Theory, Vol. 19, No. 7, 27.03.2023, p. 1671-1680.
Research output: Contribution to journal › Article › Research › peer review
}
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 -