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On the µ-invariants of abelian varieties over function fields of positive characteristic

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.

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Official URL: https://doi.org/10.2140/ant.2021.15.863


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

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