Lai, K-F and Longhi, I and Suzuki, T and Tan, K-S and Trihan, F (2021) On the µ-invariants of abelian varieties over function fields of positive characteristic. In: Algebra and Number Theory, 15 (4). pp. 863-907.
|
PDF
Alg_Numb_Theory_15_2021.pdf - Published Version Download (1MB) | Preview |
Abstract
Let A be an abelian variety over a global function field K of characteristic p. We study the µ-invariant appearing in the Iwasawa theory of A over the unramified �p-extension of K. Ulmer suggests that this invariant is equal to what he calls the dimension of the Tate�Shafarevich group of A and that it is indeed the dimension of some canonically defined group scheme. Our first result is to verify his suggestions. He also gives a formula for the dimension of the Tate�Shafarevich group (which is now the µ-invariant) in terms of other quantities including the Faltings height of A and Frobenius slopes of the numerator of the Hasse�Weil L-function of A/K assuming the conjectural Birch�Swinnerton-Dyer formula. Our next result is to prove this µ-invariant formula unconditionally for Jacobians and for semistable abelian varieties. Finally, we show that the �µ = 0� locus of the moduli of isomorphism classes of minimal elliptic surfaces endowed with a section and with fixed large enough Euler characteristic is a dense open subset. © 2021 Mathematical Sciences Publishers.
Item Type: | Journal Article |
---|---|
Publication: | Algebra and Number Theory |
Publisher: | Mathematical Science Publishers |
Additional Information: | The copyright for this article belongs to Authors |
Department/Centre: | Division of Physical & Mathematical Sciences > Mathematics |
Date Deposited: | 06 Aug 2021 08:17 |
Last Modified: | 06 Aug 2021 08:17 |
URI: | http://eprints.iisc.ac.in/id/eprint/69037 |
Actions (login required)
View Item |