Euler characteristic and Akashi series for Selmer groups over global function fields (Q1668853)

Let \(p\) be a prime and let \(A\) be an abelian variety defined over a global function field \(F\) of characteristic \(p\). Let \(K/F\) be a \(p\)-adic Lie extension of dimension \(d \geq 1\) with Galois group \(G\) such that \(G\) has no \(p\)-torsion. Under certain hypotheses, the authors provide formulae for the (truncated) Euler characteristic of the \(p\)-part of the Selmer group of \(A\) over \(K\). As usual in this context, the Selmer group is defined using flat cohomology, and the Euler characteristic of a (left) \(G\)-module \(M\) is defined to be \[ \chi(G,M) := \prod_{i=0}^{\infty} |H^i(G,M)|^{(-1)^i} \] whenever the product on the right-hand side makes sense. The terminology of truncated Euler characteristic has been introduced in [\textit{J. Coates} et al., Doc. Math. Extra Vol., 187--215 (2003; Zbl 1142.11366)] and just means that one takes the product from \(i=0\) to \(1\) only. If \(A\) is a constant ordinary variety and \(G \simeq \mathbb Z_p^d\), they deduce from work of Lai, Longhi, Tan and Trihan [\textit{K. F. Lai} et al., Proc. Lond. Math. Soc. (3) 112, No. 6, 1040--1058 (2016; Zbl 1347.11078)] that the Euler characteristic of the dual Selmer group is related to \(p\)-adic \(L\)-functions.