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On the $\mu$-invariants
of abelian varieties over function fields of positive
characteristic
King-Fai Lai, Ignazio Longhi, Takashi Suzuki, Ki-Seng Tan and Fabien Trihan |
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Vol. 15 (2021), No. 4, 863–907
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Abstract |
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Let be an abelian variety over a global function field of characteristic . We study the -invariant appearing in the Iwasawa theory of over the unramified -extension of . Ulmer suggests that this invariant is equal to what he calls the dimension of the Tate–Shafarevich group of 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 -invariant) in terms of other quantities including the Faltings height of and Frobenius slopes of the numerator of the Hasse–Weil -function of assuming the conjectural Birch–Swinnerton-Dyer formula. Our next result is to prove this -invariant formula unconditionally for Jacobians and for semistable abelian varieties. Finally, we show that the “” 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. |
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Keywords
Iwasawa theory, abelian variety, Selmer group,
$\mu$-invariant, elliptic surface
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Mathematical Subject Classification 2010
Primary: 11R23
Secondary: 11G10, 11S40, 14J27
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Milestones
Received: 3 October 2019
Revised: 2 August 2020
Accepted: 12 October 2020
Published: 29 May 2021
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Authors |
| King-Fai Lai | |
| School of Mathematics and
Statistics Henan University Henan China |
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| Ignazio Longhi | |
| Department of Mathematics Indian Institute of Science Bangalore India |
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| Takashi Suzuki | |
| Department of Mathematics Chuo University Tokyo Japan |
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| Ki-Seng Tan | |
| Department of Mathematics National Taiwan University Taipei Taiwan |
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| Fabien Trihan | |
| Department of Information and
Communication Sciences Sophia University Tokyo Japan |
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