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

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 ‫ޚ‬ 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. 1. Introduction 863 2. The µ-invariant for abelian varieties 869 3. The µ-invariant for Jacobians 879 4. The µ-invariant for semistable abelian varieties 886 5. The µ-invariant for elliptic curves 889 6. µ = 0 for generic elliptic curves 893 7. Computations 898 Appendix: Integrality of sum of slopes less than one 902 Acknowledgments 904 References 904

The $μ$-invariant change for abelian varieties over finite $p$-extensions of global fields

“…Verifying Theorem A. Using the formula (20) of [LLSTT21], we can find an equation of an elliptic curve whose associated µ-invariant equals 1 . This is the case, for example, for the curve…”

“…In this article, we discuss only the change of µ-invariants of A and we do this in a greater generality than the setting of [LLSTT21]. In particular, the first part of this article deals with the general situation of an abelian variety A over a global field (whose characteristic is not necessarily positive) and in the second part, we will specialize in the case where char(K) = p and A is an ordinary elliptic curve having semi-stable reduction everywhere.…”

“…In the work [LLTT15] of Lai, Longhi, the first and second authors, explicit examples for which µ L/K is infinite are constructed for a dihedral extension by using a multiplicative non-split place that ramifies in the extension. More recently, in [LLSTT21], the same authors together with Suzuki study the µ-invariant in the case of the unramified Z p -extension. To be more precise, they have shown that the µ-invariant along the unramified Z p -extension K (∞) p…”