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

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.

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