Iwasawa theory of \(\mathbb{Z}_p^d\)-extensions over global function fields (Q1379597)

The authors give an exposition on Iwasawa theory of \(\mathbb{Z}^d_p\)-extensions of global function fields over finite fields of characteristic \(p\), including some new results. The paper is well written and gives major algebraic results on this topic with proofs or references. They also explain the comparison to the number field case. The paper is constructed as follows. Section 2 is devoted to Iwasawa invariants of \(\mathbb{Z}_p\)-extensions under the assumption that only finitely many primes are ramified there. In section 3, they give examples of geometric \(\mathbb{Z}_p\)-extensions with arbitrarily large \(\mu\)-invariants, using the same idea as in the number field case due to K. Iwasawa. Section 4 provides some general results of \(\mathbb{Z}_p^d\)-extensions due to A. Cuoco, P. Monsky and R. Greenberg. Finally, they treat in section 5 the behavior of Iwasawa invariants in \(\mathbb{Z}^2_p\)-extensions. Let \(k_\infty/k\) be a geometric \(\mathbb{Z}_p\)-extension with constant field \(\mathbb{F}_q\) and with finitely many ramified primes. Let \(F_n\) be the extension of \(\mathbb{F}_q\) of degree \(p^n\) and put \(\lambda_n=\lambda (k_\infty F_n/kF_n)\) and \(\mu_n=\mu (k_\infty F_n/kF_n)\). The authors discuss how \(\lambda_n\) and \(\mu_n\) change as \(n\) increases and show that there exists some integer \(c\) such that \(\lambda_n= \lambda (k_\infty/k)\) and \(\mu_n= \mu(k_\infty/k) +nc\) for every \(n\) sufficiently large.