Vol. 15, No. 4, 2021

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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

Vol. 15 (2021), No. 4, 863–907
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 AK 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.

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Keywords
Iwasawa theory, abelian variety, Selmer group, $\mu$-invariant, elliptic surface
Mathematical Subject Classification 2010
Primary: 11R23
Secondary: 11G10, 11S40, 14J27
Milestones
Received: 3 October 2019
Revised: 2 August 2020
Accepted: 12 October 2020
Published: 29 May 2021
Authors
King-Fai Lai
School of Mathematics and Statistics
Henan University
Henan
China
Ignazio Longhi
Department of Mathematics
Indian Institute of Science
Bangalore
India
Takashi Suzuki
Department of Mathematics
Chuo University
Tokyo
Japan
Ki-Seng Tan
Department of Mathematics
National Taiwan University
Taipei
Taiwan
Fabien Trihan
Department of Information and Communication Sciences
Sophia University
Tokyo
Japan