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We extend the work of Lai, Longhi, Suzuki, the first two authors and study the change of $\mu$-invariants, with respect to a finite Galois p-extension $K'/K$, of an ordinary abelian variety $A$ over a $\mathbb{Z}_p^d$-extension of global fields $L/K$ that ramifies at a finite number of places at which $A$ has ordinary reductions. In characteristic $p>0$, we obtain an explicit bound for the size $\delta_v$ of the local Galois cohomology of the Mordell-Weil group of $A$ with respect to a $p$-extension ramified at a supersingular place $v$. Next, in all characteristics, we describe the asymptotic growth of $\delta_v$ along a multiple $\mathbb{Z}_p$-extension $L/K$ and provide a lower bound for the change of $\mu$-invariants of $A$ from the tower $L/K$ to the tower $LK'/K'$. Finally, we present numerical evidence supporting these results., Comment: v3, 38 pages, basically identical to v2, with clarifications regarding certain citations from [LLSTT21]

For an elliptic curve A defined over a global function field K of characteristic p>0, the p-Selmer group of the Frobenius twist of A tends to have larger order than that of A. The aim of this note is to discuss this phenomenon., Comment: 25 pages, 0 figures

Let $A$ be an abelian variety over a global function field $K$ of characteristic $p$. We study the $\mu$-invariant appearing in the Iwasawa theory of $A$ over the unramified $\mathbb{Z}_p$-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 $\mu$-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 $\mu$-invariant formula unconditionally for Jacobians and for semistable abelian varieties. Finally, we show that the "$\mu=0$" 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., Comment: Accepted for publication in Algebra & Number Theory. No changes in the text from v3. 47 pages

We generalize Iwasawa's theorem on class group over $\Z_p$-extensions to all $\Z_p^d$-extensions., Comment: 17 pages, 2 figures

We prove the Iwasawa main conjecture over the arithmetic $\mathbb{Z}_p$-extension for semistable abelian varieties over function fields of characteristic $p>0$., Comment: arXiv admin note: substantial text overlap with arXiv:1205.5945

We prove a functional equation for two projective systems of finite abelian $p$-groups, $\{\fa_n\}$ and $\{\fb_n\}$, endowed with an action of $\ZZ_p^d$ such that $\fa_n$ can be identified with the Pontryagin dual of $\fb_n$ for all $n$. Let $K$ be a global field. Let $L$ be a $\ZZ_p^d$-extension of $K$ ($d\geq 1$), unramified outside a finite set of places. Let $A$ be an abelian variety over $K$. We prove an algebraic functional equation for the Pontryagin dual of the Selmer group of $A$., Comment: 30 pages. arXiv admin note: substantial text overlap with arXiv:1205.5945

We study a geometric analogue of the Iwasawa Main Conjecture for constant ordinary abelian varieties over $\ZZ_p^d$-extensions of function fields ramifying at a finite set of places., Comment: 19 pages. arXiv admin note: substantial text overlap with arXiv:1205.5945

We study a geometric analogue of the Iwasawa Main Conjecture for abelian varieties in the two following cases: constant ordinary abelian varieties over $Z_p^d$-extensions of function fields ($d\geq 1$) ramified at a finite set of places, and semistable abelian varieties over the arithmetic $Z_p$-extension of a function field. One of the tools we use in our proof is a pseudo-isomorphism relating the duals of the Selmer groups of $A$ and its dual abelian variety $A^t$. This holds as well over number fields and is a consequence of a quite general algebraic functional equation., Comment: 80 pages; many relevant changes all over the paper from v1. Among the most significant ones: new introduction; proof of the functional equation for Gamma systems in more cases and some applications to CM abelian varieties

Consider an abelian variety $A$ defined over a global field $K$ and let $L/K$ be a $\Z_p^d$-extension, unramified outside a finite set of places of $K$, with $\Gal(L/K)=\Gamma$. Let $\Lambda(\Gamma):=\Z_p[[\Gamma]]$ denote the Iwasawa algebra. In this paper, we study how the characteristic ideal of the $\Lambda(\Gamma)$-module $X_L$, the dual $p$-primary Selmer group, varies when $L/K$ is replaced by a intermediate $\Z_p^e$-extension., Comment: 33 pages

Let K be a global function field of positive characteristic p and let M be a (commutative) finite and flat K-group scheme. We show that the kernel of the canonical localization map H^{1}(K,M)\to\prod_{all v}H^{1}(K_{v},M) in flat (fppf) cohomology can be computed solely in terms of Galois cohomology. We then give applications to the case where M is the kernel of multiplication by p^{m} on an abelian variety defined over K., Comment: A "Concluding Remarks" Section added

We establish a generalized Cassels-Tate dual exact sequence for 1-motives over global fields. We thereby extend the main theorem of [4] from abelian varieties to arbitrary 1-motives.

We prove control theorems for abelian varieties over function fields., Comment: 13 pages

We establish formulae of Stark type for the Stickelberger elements in the function field setting. Our result generalizes a work of Hayes and a conjecture of Gross. It is used to deduce a $p$-adic version of Rubin-Stark Conjecture and Burns Conjecture., Comment: 29 pages

Let $k_{\infty}$ be a $\Z_p^d$-extension of a global function field $k$ of characteristic $p$. Let $\Cl_{k_{\infty},p}$ be the $p$ completion of the class group of $k_{\infty}$. We prove that the characteristic ideal of the Galois module $\Cl_{k_{\infty},p}$ is generated by the Stickelberger element of Gross which calculates the special values of $L$ functions., Comment: 26 pages

In this note, we give a short proof of the existence of certain abelian extension over a given global field $K$. This result implies that for every positive integer $m$, there exists an abelian extension $L/K$ of exponent $m$ such that the $m$-torsion subgroup of $\Br(K)$ equals $\Br(L/K)$., Comment: 5 pages

We extend the well-known Cassels-Tate dual exact sequence for abelian varieties A over global fields K in two directions: we treat the p-primary component in the function field case, where p is the characteristic of K, and we dispense with the hypothesis that the Tate-Shafarevich group of A is finite., Comment: 9 pages, 11 references